LIBRARY OF CONGRESS. 

Shelf.. .^.S* 



UNITED ^lATES OF AMERICA. 



\ 



TABLE BOOK AND TEST PROBLEMS 



IN 



MATHEMATICS 



BY 



.A 



J. "Kr ELL WOOD, A.M. 

Principal of the Colfax School, Pittsburgh, Penn, 







NEW YORK • : . CINCINNATI • ; • CHICAGO 

AMERICAN BOOK COMPANY 



Copyright, 1892, 
By AMERICAN BOOK COMPANY, 






CONTENTS. 



"•<>♦- 



Part Page 

I. THEOREMS, RULES, AND FORMULAS 9 

Table of Logarithms of Numbers from 1 to 10,000 ... 27 
Table of Logarithmic Sines, Cosines, Tangents, and Co- 
tangents 43 

Table of Natural Sines, Cosines, Tangents, and Cotangents 89 

II. TEST PROBLEMS Ill 

Arithmetical Problems Ill 

Denominate Numbers Ill 

Least Common Multiple and Greatest Common Divisor . . Ill 

Partnership . 112 

Proportion 112 

Profit and Loss 113 

Stocks and Bonds 115 

Interest 116 

Discount and Present Worth 117 

Involution and Evolution 117 

Alligation ....>..., 118 

Annuities 118 

"Age" Problems 118 

" Time " Problems 119 

General Analysis 120 

Mensuration 122 

The Rectangle 122 

The Triangle 123 

The Circle 125 

Pyramids and Cones 125 

Q 



4 CONTENTS. 

Part Page 

Similar Solids 126 

Cubes and Spheres 127 

Miscellaneous Problems 127 

Algebraic Problems 130 

Factoring 130 

Fractions 130 

Simple Equations 131 

Radicals 133 

Quadratic Equations ^ . . 133 

Special Expedients 137 

Simultaneous Equations 137 

Reciprocal or Recurring Equations 139 

Higher Equations 140 

Miscellaneous Problems 144 

Applications of Algebra 144 

Geometrical Problems, etc 146 

Trigonometrical Problems 148 

Problems involving Calculus 149 

Promiscuous Problems 149 

Problems with Curious Besults 151 

Digits 151 

''One Cent '^ 151 

Involution of Imaginary Quantities 151 

The Zero Factor 151 

Something to investigate 151 

The Proposition of Archimedes 152 

1888 152 

Summation by Subtraction 152 

Series 153 

III. SOLUTIONS 155 



PREFACE. 



Ix nearly every class in mathematics there is to be found a larger 
or smaller number of pupils who early develop more than that 
average proficiency for which the problems in the regular text-books 
are graded and adapted. These problems do not afford sufficiently 
stimulating exercise to such pupils, whose development is often 
arrested or retarded in consequence. 

This volume of " Test Problems " has been prepared for the use 
of such apt or advanced pupils, and for the convenience of teachers 
in examining advanced classes. It will also afford valuable supple- 
mental work in every school. It contains a collection of rather 
difficult problems in the various branches of elementary mathematics. 
While none of the problems involve higher mathematics, their solu- 
tion requires close reasoning and a thorough knowledge of ele- 
mentary principles. It is believed they will afford the drill needed 
by advanced classes or by pupils of rather more than average apti- 
tude in mathematics. 

The problems have been gathered from many sources. A few of 
them may occur in the regular text-books ; many have appeared in 
the columns of mathematical or educational journals ; while still 
others have been supplied to the author by mathematicians, or are 
original w^ith himself. 

For convenient reference; there has been embodied in Part I. a 
collection of rules of mensuration, important theorems, trigonomet- 
rical formulas, and tables of logarithms and of natural sines, cosines, 
tangents, etc., which it is believed will commend itself to all. In Part 
II. is embraced the statement of the test problems, wfiich have been 
classified as closely as seemed possible. In Part III. are given the 
solutions of all problems stated in Part II. 

The solutions are not all original. Most of those that are not, 
however, are credited to their authors ; but some of them have been 
picked up "by the wayside," and their authors cannot be given. In 

5 



6 PREFACE. 

nearly all solutions the aim has been to make every step clear rather 
than to present a brief operation ; and it is believed that any student 
with a fair knowledge of any of the branches of mathematics under 
which a given problem falls will be able to follow its solution intelli- 
gently and easily. The arithmetical and algebraic solutions are 
designed especially to aid the pupils and teachers of our public 
schools; while the chapters on *' Special Expedients" and "Miscel- 
laneous Solutions " contain much that may be studied with profit by 
more advanced scholars, teachers, superintendents, etc., as they include 
solutions by some of the best mathematicians of the country. In 
solving equations, it has not been deemed necessary to give the values 
of all the unknown quantities ; yet in some instances they have been 
given, and in the others they are readily obtainable. 

The classification of the solutions is necessarily imperfect, owing to 
their promiscuous character, and to the fact that frequently two or 
more principles are almost equally involved in the solution of a 
problem, on account of which the solution might as legitimately be 
placed in another class as in the one to which it has been assigned. 
Especially is it difficult to classify " Special Expedients," or ** Artifices," 
as they are restricted and special, and as in their use the student must 
depend solely upon his ingenuity. Owing to these difficulties, the 
classification throughout has been based upon the topic or head under 
which a solution chiefly falls, or, in other words, upon the leading- 
principle or operation involved. 

For fine solutions received, I desire to express my thanks to Dr. 
I. J. AVireback and Mr. L. B. Fillman of St. Petersburgh, Penn., and 
to Professor B. F. Burleson of Oneida Castle, N.Y., who has also 
rendered valuable assistance in reading proof, in removing imper- 
fections, and in supplying many of the problems in " Series," together 
with their admirable solutions. Special acknowdedgment is due to 
Mr. Russell Hinman of the American Book Company, New York, to 
whom I am indebted for many excellent suggestions and uniform 
courtesy. 

J. K. E. 



INDEX TO EULES AND THEOREMS, 



Beams. See Timbers. 

Bushel, cubic contents of . . . 17 

Circle, area of 1 

area of a sector of 2 

area of a segment of ... . 3 

chord of an arc and half arc of 4 

circumference of 5 

and equivalent square .... 6 

Circles compared 7 

inscribed 61 

Cone. See Pyramid. 

Cosines, table of natural . . . . - 

table of logarithmic - 

Cotangents, table of natural . . - 

table of logarithmic - 

Cube, diagonal of 8 

Cubes, inscribed 9 

Cycloid, area of 10 

Cycloidal curve, length of ... 11 
Cylinder. See Prism. 
Difference and sum of two quan- 
tities, relations of 17 

Earth, radius of 17 

Ellipse, area of 12 

circumference of 13 

Ellipsoid, surface of 14 

Gallon, cubical contents of . . . ,17 

Helix, length of 15 

Kilogram, value of French, in 

pounds 17 

Liter, value of French, in cu. in. 17 

Logarithms, table of 

Lune, area of 16 

Meter, value of French, in feet . 17 

Miscellaneous formulas .... 17 

trigonometrical formulas . . 73 

TT, value of 17 

value of Log 17 

Parabola, area of 18 

area of a segment of .... 19 



age 


Paraboloid, volume of ... . 


No. 
20 


Page 
14 


12 
9 


Pendulum, length of seconds, at 
New York, London, and 






q 


Paris 


17 
21 

22 


12 


9 
9 


Polygon, area of a regular . . . 
surface of a spherical .... 


14 
14 


10 


Prism, surface of 


23 


14 


10 


volume of 


24 


14 


10 


Prismoid, volume of . . . . . 


25 


15 


20 


Pyramid or cone, center of grav- 






89 


ity of a triangular 

volume of 


26 

27 


15 
15 


43 


volume of a frustum of . . . 


28 


15 


99 
43 


Ring, length of axis of an ellipti- 
cal 


29 
30 


15 


10 


volume of a cylindrical . . . 


15 


10 


Sines, table of natural .... 


_ 


89 


11 
12 


table of logarithmic .... 
Sphere, surface of 


31 


43 
15 




surface of a segment of . . . 
volume of 


32 
33 


15 
15 


13 
13 


volume of a segment of . . . 
Spheres, inscribed 


34 
9 


15 

10 


12 


Spherical sector, volume of. . . 


35 


16 


12 
12 


Spheroid, surface of 

surface of a frustum of . . . 


36 
37 


16 
16 


12 

12 


surface of a segment of . . . 
volume of 


38 
39 


16 
16 




volume of a frustum of . .40 


,41 


16 


12 

12 


volume of a segment of . .42 
Spindle, surface of a circular . . 


,43 
44 


17 
17 


27 
12 


surface of a cycloidal .... 
volume of a circular .... 


45 

46 


17 
17 


12 
12 


volume of a segment of a cir- 
cular 


47 


17 


24 


volume of a zone of a circular . 


48 


18 


12 
12 
14 


volume of a cycloidal .... 
volume of an elliptical. . . . 
volume of middle frustum of an 


49 
50 


18 
18 


14 


elliptical 


61 


18 



8 



INDEX TO RULES AND THEOBSMS. 



Spindle (continued). 
volume of segment of an ellip 

tical 

volume of a parabolic . . . 
volume of middle frustum of 

a parabolic 

volume of segment of a para 

bolic 

volume of an hyperbolic . . 
volume of middle frustum of 

an hyperbolic 

volume of segment of an hj^per 

bolic 

Spiral line, length of a plane . 
Square, circle an(} equivalent , 
equilateral triangle and equiv 

alent 

Square of any number . . . 
Squares, inscribed .... 61 
Sum and difference of two quan 

titles, relations of. . . . 
Tangents, table of natural . . 



No. Page 

52 18 

53 18 

54 18 

55 18 

56 19 

57 19 



58 


19 


59 


19 


6 


10 


68 


22 


60 


19 


2 20 


21 



17 13 



No. Page 
Tangents (continued). 

table of logarithmic - 43 

Timbers, strength of 63 21 

Trapezium, area of 64 22 

Trapezoid, area of 65 22 

center of gravity of 66 22 

Triangle, area of 67 22 

area of equilateral, equivalent 

to a given square 68 22 

center of gravity of 69 22 

Triangles compared 70 22 

formulas for oblique .... 71 23 

formulas for right-angled. . . 72 23 
Trigonometrical formulas, mis- 

cellaneous 73 24 

Ungulas, curved surface and vol- 
ume of 74 25 

Water, weight of cubic foot of . 17 12 

Wedge, volume of 75 26 

Zone, area of a circular .... 76 26 

surface of a spherical .... 77 26 

volume of a spherical .... 78 26 



TABLE BOOK AND TEST PROBLEMS 



IN 



MATHEMATICS 



-ooXKc 



Part L 
theorems, rules, and formulas. 

1. Area of a Circle. — Multiply the square of the radius by 
TT, or 3.1416, or use the formula Trr^. 

2. Area of a Sector of a Circle. — Multiply the arc by half 
the radius; or use this proportion, Area of circle : area of sec- 
tor : : 360 : number of degrees in arc. 

3. Area of a Segment of a Circle. — Find the area of a sector 
having the same arc, and the area of the triangle formed by 
chord of segment and radii of sector. The difference of these 
is the area of a segment less, and their sum that of one greater, 
than a semicircle. Or (^approximate rule when segment is less 
than semicircle') to two thirds of the product of height of seg- 
ment by chord, add cube of height divided by twice chord. 

4. The Chord of an Arc. — Let (7= the chord of an arc, 
c = the chord of half an arc, v sin = versed sine, d = diam- 
eter. Then 

■|(8 c — G) = length of arc nearly. 

V(7^ + 4:V sin^ X 10 V sin^ , ^ , .v p 

15C- + 33vsin- + 2 c = length of arc. 

9 



10 TABLE BOOK AND TEST PROBLEMS. 



length of arc. 



2cx 


10 V sin 


■ + 2c 


60d- 


-27vsin 




2Vc2~ 


V sin^ 



a 

Vd2-(fZ-vsinx2)-^=a 
i((72 + 4vsm2)* = c. 
-\/d X V sin = c. 
c^ -^ V sin = d. 
c^ -i- d = Y sin. 



i(c|_ Vc^'-C') = vsin. 
When V sin is greater than a radius, 



-i(d+V^'-C-) = vsin. 

5. Circumference of a Circle. — Multiply the product of the 
radius and 3.1416 by 2, or use the formula 27rr, 

6. Circle and Equivalent Square. — Diameter of circle multi- 
plied by .8862 is equal to the side of an equal square. 

7. Circles are to each other as the squares on their radii. 

8. Given the Side of a Cube, to find its Diagonal. — The 

diagonal of the side is V2 x side^. This is one leg, and the 
edge of the cube is another leg, of a right-angled triangle 
whose hypothenuse is the diagonal of the cube. Hence the 
cube's diagonal = VS x side^ = side x V3. Therefore, to find 
the diagonal of a cube from its side, multiply the side by V3. 

9. Inscribed Cubes and Spheres. — A cube is inscribed in a 
sphere, a sphere in this cube, a cube in this sphere, and so on. 
Find the ratio of the first sphere to the fifth. 

The diameter of the first sphere is readily seen to be the 
diagonal of the first cube, and the edge of the first cube to be 
the diameter of the second sphere, and so on. If a = the edge 
of the cube, then a' + a- -\-a=3a^ = square of diagonal, and 
aV3 = diagonal. Hence, to find the edge of a cube, divide the 
diagonal by V3. 



THEOREMS, BULES, AND FORMULAS, 11 

Let i? = radius of first sphere. Then 2i^ = its diameter = 

— 2 7? 

diagonal of first cube. *2R-^ V3 = — -^= edge of first cube = 

^ 2R 

diameter of second sphere = diagonal of second cube. — ^ -j- 

2R "^^ 

VS = = edge of second cube = diameter of third sphere 

o 

= diagonal of third cube, and so on ad infinitum. 
We now observe that the first three diameters are 2B,, 

2 1^ 2 R 2 J^ ^ Tf 

J , the diagonals the same, and the edges —~, — ^, 

V3 3 V3 3 

~, each forming part of an infinite decreasing series, whose 

3V3 

ratio is — -* 

Having given the diameter of the first sphere, we may find 
any diameter by the rule or formula for finding the last term 
of a geometrical progression. Let R = radius, 2R = diameter 
of the first sphere. Then, in the problem, 



.VSJ 9 

2 7? 

is the fifth term of the series, or the diameter of the fifth 

9 

sphere. By similar solids, we have, First sphere : fifth sphere 

::2R^ : f j ; that is, as 2 R^ : — — -. Hence the first sphere 

is 729 times the fifth. 

2 7? 2 7? 

As the diameters form the following series, 2 R, — -, ? 

a/,S 3 
27?27?27?27? 

-^ - — , -, , etc., we observe that any desired diam- 

3V3 9 9V3 2^ 

eter may be found by dividing the first diameter by (V3)""^ 

2 R 2 R 

For example, the seventh diameter = = 

10. Area of a Cycloid. — Multiply the area of the generat- 
ing circle by 3. 



12 TABLE BOOK AND TEST PROBLEMS, 

11. Length of a Cycloidal Curve. — Multiply the diameter 
of the generating circle by 4. 

12. Area of an Ellipse. — Multiply the product of the diam- 
eters by ^TT. 

13. Circumference of an Ellipse. — Let D and d represent 
the long and short diameters. Then the circumference equals 



TT 



-y/ — ^t i L {^approximate) 



14. Convex Surface of an Ellipsoid. — To four times the 
square of the height add the square of the base. Multiply 
the square root of half the sum by 3.1416, and this product by 
the radius of the base. 

15. Length of a Helix. — Square the circumference de- 
scribed by the generating point, add the square of the distance 
advanced in one revolution, extract the square root of the sum, 
and multiply by the number of revolutions. 

16. Area of a Lune or Crescent. — Take the difference of the 
areas of the two segments formed by the arcs of the lune and 
its chord. 

17. Miscellaneous Formulas. 

TT r= 3.14159 26535 89793 23846 26433 83280. 
Log TT = 0.49714 98726 94133 85435 12682 88291. 
United States standard gallon = 231 cu. in. = 0.133681 cu. ft. 

United States standard bushel == 2150.42 cu. in. = 1.244456 cu. ft. 
British imperial gallon = 277.25678 cu. in. = 0.160449 cu. ft. 

French meter = 3.28083 ft. 
French liter = 61.02327 cu. in. 
French kilogram = 2.20462 lbs. Avoirdupois. 
Weight of cubic foot of water (maximum density 39.101°, barometer 

30 in., thermometer 39.83° F.) = 62.379 lbs. Avoirdupois. 
Weight of cubic foot of water (maximum density 39.101°, barometer 

30 in., thermometer 62° F.) = 62.321 lbs. Avoirdupois. 
Length of seconds pendulum at New York = 39.10120 in. 
Length of seconds pendulum at London = 39.13908 in. 
Length of seconds pendulum at Paris = 39.12843 in. 



THEOREMS, RULES, AND FORMULAS. 13 

Equatorial radius of earth according to Clarke — 20926061.779 ft. 
Polar radius of earth according to Clarke = 20855120.854 ft. 

Mean radius of earth = 20890591.316 ft. 

Calling the earth a sphere with the above mean radius, 100 feet on 

the surface subtends an angle of .987355124 of a second at the 

center of the earth. 

Let s = the surn^ d = the difference, p = the product, of two 
numbers A and B, of which A is the larger number. Then we 
have the following formulas : — 

One half the sum of two quantities plus one half their differ- 
ence is equal to the larger quantity, or ^s + ^d = A. 

One half the sum of two quantities minus one half their 
difference is equal to the smaller quantity, or ^s — ^d = B. 

The square root of the difference between the square of the 
sum of two quantities and four times their product is equal to 
the difference of the quantities, or Vs'-^ — 4:p = A — B, 

The square root of the sum of the square of the difference of 
two quantities and four times their product is equal to the 
sum of the quantities, or VcZ^ + 4:p = A + B. 

The difference of two quantities divided by one less than 
the quotient arising from dividing the larger by the smaller 

is equal to the smaller quantity, or d-^ ( l\ = B. 

The square of the sum of two quantities minus the sum of 
their squares is equal to twice their product, or s^ — {A^ + B'^) 
=^2 p. 

The square root of the quotient obtained by dividing the 
product of two quantities by the quotient of the larger by the 

smaller is equal to the smaller quantity, or -xlp h — = B. 

The difference of the squares of two quantities divided by 
the sum of the quantities is equal to the difference of the 
quantities, or {A^ — B^) -^ s = d. 



14 TABLE BOOK AND TEST PROBLEMS. 

The square of the sum of two quantities is equal to the 
square of the first, plus the square of the second, plus twice 
the jjroduct of the two, or s^ = A'^ + B^ + 2p. 

The square of the difference of two quantities is equal to the 
square of the first, jd^^^s the square of the second, minus twice 
the product of the two, or d- = A' + jB- — 2p. 

The product of the sum and difference of two quantities is 
equal to the difference of their squares, or sd = A'^ — B^. 

The chord of an angle is equal to twice the sine of half the 
angle. 

18. Area of a Parabola. — Take two thirds of the product of 
the base by the height.* 

19. Area of a Segment of a Parabola. — Multiply the differ- 
ence of the cubes of the two ends of the segment by twice its 
height, and divide the product by three times the difference 
of the squares of the ends. 

20. Volume of a Paraboloid. — Multiply the product of the 
height and the square of the radius by tt, and divide the result 
by 2. 

21. Area of a Regular Polygon. — Use the formula 

^a^n cot (a = side, n = number of sides). 

n 

22. Surface of a Spherical Polygon. — Use the formula 

ttv^ X — >~ ^^^ "~ — (r = radius of sphere, S = sum of an- 

180° ^ ^ 

gles, n = number of sides). 

23. Surface of a Prism or Cylinder. — Multiply the perim- 
eter by the height, and add the areas of the two ends. 

24. Volume of a Prism or Cylinder. — Multiply the area of 
the base by the height. 

* The area of a circular Koijment on railroad curves, wliere the chord is very long in 
proportion lo the heiglit, may be found with great accuracy by the above formula. 



THEOREMS, RULES, AND FORMULAS, 15 

25. Volume of a Prismoid. — Use the Prismoidal Formula : 
Add the areas of the two bases to four times the area of a 
middle section parallel to them, and multiply the sum by one 
sixth of the perpendicular height. 

26. The Center of Gravity of a Triangular Pyramid is in the 
line joining the vertex and the center of gravity of the base, 
at one fourth the distance from the base to the vertex. 

27. Volume of a Pyramid or Cone. — Multiply the area of the 
base by one third of the altitude. 

28. Volume of a Frustum of a Pyramid or Cone. — Multiply 
the areas of the two bases together, and extract the square 
root of the product. To this root add the two areas, and mul- 
tiply the sum by one third of the altitude. 

29. Length of Axis of an Elliptical Ring. — Square the 
diameters of the axes of the ring, and multiply the square 
root of half their sum by 3.1416. 

30. Volume of a Cylindrical Ring. — Multiply the sum of 
the thickness and inner diameter by the square of the thick- 
ness, and that product by 2.4674. 

31. Surface of a Sphere. — Multiply the diameter by the cir- 
cumference. 

32. Surface of a Segment of a Sphere. — Multiply the height 
by the circumference of the sphere, and add the area of the 
base. 

33. Volume of a Sphere. — Multiply the cube of the diameter 
by .5236. 

34. Volume of a Segment of a Sphere. — Add the square of 
the height to three times the square of the radius of the base, 
and multiply the sum by the product of the height by .5236 ; 
or subtract twice the height of the segment from three times 
the diameter of the sphere, and multiply the remainder by the 
product of the square of the height by .5236. 



16 TABLE BOOK AND TEST PROBLEMS, 

35. Volume of a Spherical Sector. — Multiply one third of 
the radius of the sphere by the external surface of the zone, 
which is the base of the sector. 

36. Surface of a Spheroid.^ — Multiply the square root of 
half the sum of the squares of the diameters by 3.1416, and 
this product by the conjugate diameter if prolate, and by the 
transverse if oblate. 

37. Convex Surface of a Frustum of a Spheroid. — Proceed 
as by the following rule to obtain proportionate height of 
frustum. Then multiply this height by 3.1416, and this prod- 
uct by the diameter parallel to the base of the frustum. 

38. Convex Surface of a Segment of a Spheroid. — Add the 

squares of the diameters, and take the square root of half 
the sum. Then as the diameter from which the segment is 
cut is to this root, so is the height of the segment to the pro- 
portionate height required. Multiply the other diameter by 
3.1416, and this by the proportionate height of the segment. 

39. Volume of a Spheroid. — Multiply the square of the 
revolving axis by the fixed axis, and this product by .5236. 

40. Volume of the Middle Frustum of a Spheroid (when 
ends are circular) . — Add the square of the diameter of either 
end to twice the square of the revolving axis, and multiply 
the sum by the product of the length of frustum by .2618. 

41. Volume of the Middle Frustum of a Spheroid (when 
ends are elliptical). — Add the product of the transverse and 
conjugate diameters of either end to twice the product of the 
transverse and conjugate diameters of the middle section, and 
multiply the sum by the product of the length of the frustum 
by .2618. 



* A spheroid is a eolid generated by the revolution of a semi-ellipse about one of its 
diameters. 



THEOREMS, BULES, AND FORMULAS. 17 

42. Volume of a Segment of a Spheroid (when base is circu- 
lar). — Take the difference between three times the fixed axis 
and twice the height of the segment, and multiply the remain- 
der by the square of the height of the segment, and this 
product by .5236. Then the square of fixed axis : square of 
revolving axis : : last product : volume. 

43. Volume of a Segment of a Spheroid (when the base is 
elliptical or perpendicular to the revolving axis) . — Take the 
difference between three times the fixed axis and twice the 
height of the segment, and multiply the remainder by the square 
of the height of the segment, and this product by .5236. Then 
the fixed axis : revolving axis : : last product : volume of seg- 
ment. 

44. Convex Surface of a Circular Spindle. — Multiply the 
radius of the revolving arc by the length of the spindle. Mul- 
tiply the arc by the distance between center of spindle and 
center of revolving arc. Subtract this product from the 
former, and multiply the remainder by 27r. 

Note. — This rule gives also the surface of a zone, segment, or frus- 
tum of a spindle. 

45. Convex Surface of a Cycloidal Spindle. — Multiply the 
area of the generating circle by ^~, 



46. Volume of a Circular Spindle. — Multiply half the area 
of the revolving segment by the central distance. Subtract 
the product from one third the cube of half the length, and 
multiply the remainder by 47r. 

47. Volume of a Segment of a Circular Spindle. — From half 
the length of the spindle take the length of the segment. 
Eind the volume of a middle frustum whose length is twice 
this difference. Then from the volume of the whole spindle 
take the volume of the middle frustum, and divide the remain- 
der by 2. 

ellwood's test prob, — 2. 



18 TABLE BOOK AND TEST PROBLEMS. 

48. Volume of a Zone or Frustum of a Circular Spindle. — 

Subtract one third of the square of half the length of the 
frustum from the square of half the length of the whole spin- 
dle, and multiply the remainder by half the length of the 
frustum. From this product subtract the product of the cen- 
tral distance by the revolving area which generates the frus- 
tum, and multiply the remainder by 27r. 

49. Volume of a Cycloidal Spindle, — Square twice the 
diameter of the generating circle, multiply by 3.927 times the 
circumference, and divide the product by 8. 

50. Volume of an Elliptic Spindle. — Add the square of its 
diameter to the square of twice the diameter at one fourth of 
its length, and multiply the sum by the product of the length 
by .1309 (or 2'j7r). 

51. Volume of the Middle Frustum of an Elliptic Spin- 
dle. — Add the squares of the greatest and least diameters to 
the square of twice the diameter midway between the two, 
and multiply the sum by the product of the length by .1309. 

52. Volume of a Segment of an Elliptic Spindle. — To the 

square of the diameter of the base of the segment add the 
square of twice the diameter midway between the base and 
the vertex, and multiply the sum by the product of the length 
of the segment by .1309. 

53. Volume of a Parabolic Spindle. — Multiply the square 
of the diameter by the length, and the product by j^tt. 

54. Volume of the Middle Frustum of a Parabolic Spin- 
dle. — To eight times the square of the greatest diameter add 
three times the square of the least diameter and four times 
the product of these diameters, and multiply the sum by the 
product of the length by -g^^Tr. 

55. Volume of a Segment of a Parabolic Spindle. — To the 

square of the diameter of the base of the segment add the 



THEOREMS, RULES, AND FORMULAS. 19 

square of twice the diameter midway between tlie base and 
vertex, and multiply the sum by the product of the height of 
the segment by J^ ^• 

56. Volume of an Hyperbolic Spindle. — Add the square of 
the diameter to the square of twice the diameter at one fourth 
its length, and multiply the sum by the product of the length 

57. Volume of the Middle Frustum of an Hyperbolic Spin- 
dle. — Add the squares of the greatest and least diameters to 
the square of twice the diameter midway between the two, 
and multiply the sum by the product of the length by -^tt, 

58. Volume of a Segment of an Hyperbolic Spindle. — To 

the square of the diameter of the base of the segment add the 
square of twice the diameter midway between the base and 
vertex, and multiply the sum by the product of the length of 
the segment by -^^ir. 

59. Length of a Plane Spiral Line. — Multiply half the sum 
of the greater and less diameters by 3.1416, and again by the 
number of revolutions ; or multiply the number of revolutions 
by the mean length of the circumferences. 

60. Square of any Number (Novel Method). — The follow- 
ing method of squaring any number was first brought to our 
notice by Mr. A. L. Foote of Merrick, N.Y., who says he has 
used it in his own practice for upwards of thirty years, but 
does not claim anything original in its use, it being mei?fely an 
application of the well-known principle that the square of any 
polynomial is equal to the sum of the squares of its several 
terms, plus twice the product of every two terms of the poly- 
nomial. For example : — 

(a + b + c + ay 

= a^-\-h^ + c^ + d^ + 2ab + 2bc + 2cd + 2ac + 2bd + 2ad. 



20 



TABLE BOOK AND TEST PROBLEMS, 



Take the number 4567, and denoting 4 by a, 5hj b, 6 by c, 
and 7 by d, we have the following arrangement : 



a 


h 




c 


d 


4 


5 




6 


7 


16 


25 




36 


49 


2ab 




2 be 


2cd 





40 



2ac 
48 



60 



2 ad 
56 



84 



2bd 
70 



Sum = 20857489 = (4567)1 The placing and addition of 
these numbers are better shown by the following : — 

16253649 

406084 

4870 

56 

20857489 

This shows us that a^ = 16,000,000, not 16, and that 
6^ = 250,000, not 25 merely. Any number whatever may be 
squared by this method. 

61. Inscribed Squares and Circles. — Let a square be in- 
scribed in a circle, a circle in this square, a square in this 

circle, and so on. It will be 
observed from the figure that 
the diameter of the first circle 
is also the diagonal of the first 
square, and that the side of 
the first square is also equal 
to tlie diameter of the second 
circle. This law holds good 
ad infinitum. 

Let the side of any square, 
as AC, =a. Then 

a'-^(r=2a- = AD% 




THEOREMS, RULES, AJSD FORMULAS. 21 

and AD = a V5 = the diagonal. Hence, to find the side of a 
square, the diagonal being given, divide the diagonal by V2. 
Let D = diameter of the first circle. It is also the diagonal of 

the first square. Then — - = side of the first square = diame- 
ter of second circle. It is also the diagonal of the second 
square : hence — - -r- V2 = — = side of the second square = di- 
ameter of third circle = diagonal of third square. Then 

— ^-V2 = = side of third square = diameter of fourth 

2 ^2V2 ^ 

circle = diagonal of fourth square, and so on. 

Reviewing this, we observe that the diameters and sides 
form two decreasing series, having the same ratio, but dif- 
fering in first terms. The diameters (and dia,^onals) are 

Z>, -— , -, — -, etc. The sides are -— , -, — -, -, etc. 

V2 2 2V2 V2 2 2V2 4 

The ratio is . 

V2 

62. The Diameter of a Circle multiplied by .707107 is equal to 

the side of an inscribed square. 

63. Strength of Timbers or Beams. — A beam twice as wide 
as another is twice as strong ; one twice as deep is four times 
as strong ; while one twice as long is only half as strong. 

A beam loose at both ends and loaded in the middle will 
bear only two thirds as much as if both ends were firmly fixed. 

A beam will bear twice as much weight uniformly distrib- 
uted over its whole length as when the entire weight is placed 
in the middle* 

When a beam is fixed at one end and loaded at the end pro- 
jecting, it will bear but one fourth the weight it will support 
when fixed at both ends and loaded in the middle. 

The strength of rectangular beams varies as the breadth 
multiplied by the square of the depth. Hence a beam with 
its narrow side upward is as much stronger than with its broad 
side upward as the depth exceeds the breadth. 



22 TABLE BOOK AND TEST PROBLEMS. 

A triangular beam is twice as strong when resting on a side 
as when resting on the opposite edge. 

An inch square bar will support a greater weight than an 
inch round one. 

A piece of oak one foot long and one inch square, when sup- 
ported at both ends, will sustain a weight of 600 pounds. A 
similar bar of iron will sustain a weight of 2190 pounds. The 
oak weighs half a pound, and the iron 3 pounds. 

A beam 2 by 8, placed on its edge, is four times as strong 
as one 2 by 4. Placed on their broad sides, the former is only 
twice as strong as the latter. 

64. Area of a Trapezium. — Multiply the diagonal by half 
the sum of the two perpendiculars falling upon it from the 
opposite angles. 

65. Area of a Trapezoid. — Multiply half the sum of the 

parallel sides by the perpendicular distance between them. 

66. The Center of Gravity of a Trapezoid is on the line 
which bisects the parallel bases, and divides it in the ratio of 
twice the longer plus the shorter to twice the shorter plus the 
longer. 

67. Area of a Triangle. — Multiply the base by half the 
altitude, or use the formula, 



Area = Vs(s — a) {s — b){s — c), 

in which s = half the sum of the sides a, b, and c. When the 
triangle is equilateral, the formula becomes, 

Area = ^a^V5. 

68. Equilateral Triangle and Equivalent Square. — The side 
of a square multiplied by 1.52 is equivalent to the side of an 
equilateral triangle of equal area. 

69. The Center of Gravity of a Triangle is one third the 

distance from the middle of a side to the opposite angle. 

70. Similar Triangles are to each other as the squares on 
their homologous sides. 



THEOREMS, RULES, AND FORMULAS. 



23 



71. Formulas for Oblique Triangles. 




Given. 


Sought. 


A, B, a 


b 


A, a, h 


B 


a, h, C 


A-B 


a, 5, c 


A 


A, B, C, a 


Area 


A, h, c 


Area 


a, ?), c 


Area 



Formulas. 



b = 



sin 5 = 



asin B 
sin J. 
6 sin A 



tan ! fA-B) = (^ - ^) tan U^ + ^) . 



If s=}(« + ^ + c), sin.^^3. JIlnMlziO, 

' he 

^ ^ - he " ^ s{s^a) 



cos 



sin A = 2^g(^-"^)(^-^)(g-^) 
he 



Area 



a^ sin j5 sin C 



2 sin ^ 
Area =hhc sin A 



= 4 (a + ?> + c) , Area = Vs(s — a)(s — h) (s — c). 



72. Formulas for Right Triangles. — Let A be any acute 
angle, and let a perpendicular BC be drawn from any point 
in one side to the other side. Then, if the sides of the right 
triangle thus formed are denoted by letters, as in the accompa- 
nying figure, we shall have these six formulas : — 



.a 

1. sm A = -' 

c 

2. cosJ[=-. 

c 



3. tan A 



a 



4. cosec A = ~' 

a 

5. sec A = — 

b 

6. cot^ =-. 

a 




24 



TABLE BOOK AND TEST PROBLEMS. 



Given. 


Sought. 


a, c 


A,B,h 


a, h 


A,B,e 


A, a 


B,h,c 


A, b 


B, a,c 


A, c 


B, a, h 



Formulas. 



sin A = -^ 
c 



^ - 90" - A, 

5 :=, 90O ^ A, 



COS B = -% 
c 



cot 5 =r -, 



6 = a cot A, 

a — b tan ^, 
a = c sin J , 



6= V(c + a)(c-a), 



c= Va^ + b'\ 
a 



c = 



sin^ 

cos^ 
b = ccos^. 



73. Miscellaneous Trig^onometrical Formulas. 

sin^^ + cos^ J. = 1. B 

sin (A ± B) = sin A cos B ± sin ^ cos A. 

a 

COS (^ ± J5) = cos A cos jB q= sin ^ sin B. 

sin 2^ = 2 sin ^ cos A. 




C b A 

cos 2 J. = cos^ A — sin- ^ = 1 — 2 sin^ A = 2 cos- ^ — 1. 

sin^ A = \ — i cos 2 A cos- J. = ^ + i- cos 2 A 

sin ^ + sin i^ = 2 sin ^{A-\-B) cos i (^1 - B). 

sin^ - sin jB = 2 cos i (.4 + B) sini- (.4 - B), 

cos J. + cos 5 = 2cos I (^4 + B) cos | (yl — B), 

cos J5 - cos ^ = 2 sin i (yl + .B) sin i- (.4 - i5). 

sin^^- sin2jB = cos^i?- cos2^= sin {A + 5) sin (^ - B). 

cos- ^1 - sin^ B = cos (^ + B) cos (.4 - B) . 



tan /I = 



sin A 

cos^ 



cot ^4 = 



cos A 
sin A 



THEOREMS, BULES, AND FORMULAS. 



25 



tan {A ± B) 



tan A ±t^TiB 
1 T tan A tan B 



tan A ± tan .B = ^^ — - — - 

cos Acqs B 



cot A±cotB= ± 



sin (A±B) 



sm ^ sm B 

sin ^ + sin .5 __ tan -^ (^ 4- ^) 
sin A — sin J5 tan ^ {A — B) 

sin ^ + sin B 



cos ^4 + cos B 



tan I- (^ + 5) 



sin 


A + 


sin 


B 


cos 


B- 


cos 


A 


sin 


A- 


sin 


B 


cos 


A + 


cos 


B 


sin 


A- 


sin 


B 



cosB 



tani J. 



cos^ 

sin^ 



= cotl(A-B), 
= t2in ^{A-B), 
= cot ^(A + B), 



cot + ^ = 



1 4- cos A 

sin A 
1 — cos A 



74. Curved Surface and Volume of Cylindrical TJngulas. 

(a) When section is parallel to axis of cylinder^ 
Curved surface = height x length of arc of one end. 
Volume = area of base x height of cylinder. 

(b) When section passes obliquely through opposite sides 
of cylinder, 

Curved surface = circumference of base of cylinder x 
half sum of greatest and least heights 
of ungula. 

Volume = area base of cylinder x half sum of 

greatest and least lengths of ungula. 

(c) When section passes through base and one side, and 
base of ungula does not exceed a semicircle, 

Curved surface = — (sine of half arc of base x diam- 

vsin 

eter of cylinder — length of arc x 

cosine). 

Volume = ^^^^^^ (2 sin^ of half arc of base - 

V sin 

area of base x cosine of half arc). 



26 TABLE BOOK AND TEST PROBLEMS, 

{d) When base exceeds a semicircle, 

Curved surface = — ^ — [sine of half arc of base x diam- 
vsm 

eter of cylinder + length of arc x 

(vsin — sine of base)]. 

Volume = — ^ — (-1 sin^ of half arc of base + area 

vsm ^^ 

of base x cosine). 

(e) When section passes obliquely through both ends of 
cylinder, find the surface as follows : — 

Conceive the section to be continued till it meets the side of 
cylinder produced. Then as the difference of versed sines 
of arcs of two ends of ungula is to v sin of arc of less end, so 
is height of cylinder to part of side produced. 

The surface of each of the ungulas thus found may be ascer- 
tained by preceding rules, and their difference will be the 
curved surface. Find the volume of each ungula by preceding 
rules, and take their difference for volume. 

75. Volume of a Wedge. — Add twice the length of the base 
to the length of the edge, and multiply the sum by one sixth 
of the product of the height of wedge and breadth of base. 

76. Area of a Circular Zone. — Subtract the areas of the 
segments from the area of the circle. 

77. Surface of a Spherical Zone. — Multiply height by cir- 
cumference of the sphere, and add the area of the two ends. 

78. Volume of a Spherical Zone. — Add one third of the 
square of height of zone to sum of the squares of the radii of 
ends, and multiply the sum by product of height by 1.5708. 



TABLE 



OF 



LOGARITHMIC SINES, COSINES, 
TANGENTS, AND COTANGENTS 



FOR 



EYEEY DEGEEE AND MINUTE 

OF THE QUADRANT. 



Note. — The minutes in the left-hand column of each page, increas- 
ing downwards, belong to the degrees at the top ; and those increasing 
upwards, in the right-hand column, belong to the degrees below. 



43 



44 




(0 Degree.) a table 


OF LOGARITHMIC 






M. 1 


Sine 


1 D 


1 Cosine. 


D. 


Tang. 


D 


Cotang. 


r 





0.000000 




10.000900 




0.000000 




Infinite. 


60 


I 


6.'*63726 


501717 


000000 


00 


6.463726 


501717 


13.536-274 


59 


2 


764750 


293485 


0>W000 


00 


764756 


293483 


235244 


58 


3 


940847 


208231 


01)0000 


00 


940847 


203231 


059153 


57 


4 


7.065786 


161517 


000000 


00 


7.065786 


161517 


12.934214 


53 


5 


162695 


131968 


000000 


00 


182696 


131969 


837304 


55 


6 


241877 


111575 


9.999999 


01 


241878 


111578 


758122 


54 


1 


308824 


93653 


999999 


01 


308825 


99 5.53 


691175 


53 


8 


366816 


85254 


999999 


01 


3G6317 


85254 


633183 


52 


9 


4179S8 


76263 


999999 


01 


417970 


76253 


582030 


51 


10 


463725 


68988 


999998 


01 


463727 


63938 


536273 


50 


11 


7.505118 


62981 


9.999998 


01 


7.505120 


62981 


12.494883 


49 


12 


542906 


57936 


999997 


01 


542909 


57933 


457091 


48 


13 


577668 


53641 


999997 


01 


577672 


53642 


4223-28 


47 


14 


009853 


49938 


999996 


01 


609857 


49939 


390143 


46 


15 


639816 


46714 


999993 


01 


619820 


46715 


360180 


45 


16 


667845 


43881 


999995 


01 


667849 


43882 


332151 


44 


17 


094173 


41372 


999995 


01 


694179 


41373 


305821 


43 


18 


718997 


39135 


999994 


01 


719003 


39138 


280997 


42 


19 


742477 


37127 


999993 


01 


742484 


37123 


257516 


41 


20 


764754 


35315 


999993 


01 


764761 


35136 


235-239 


40 


21 


7.785943 


33672 


9.999992 


01 


7.785951 


33673 


12.214049 


39 


22 


806146 


32175 


999991 


01 


803155 


32176 


193845 


38 


23 


825451 


30805 


999990 


01 


825450 


30806 


174540 


37 


24 


843934 


29547 


999989 


02 


843944 


29549 


156056 


36 


25 


861662 


28388 


999988 


02 


861674 


28390 


138326 


35 


26 


878695 


27317 


999988 


02 


878703 


27318 


121292 


34 


27 


895085 


26323 


999987 


02 


895099 


263-25 


104901 


33 


28 


910879 


25399 


999985 


02 


910394 


25401 


089106 


32 


29 


926119 


24538 


999985 


02 


926134 


24540 


073866 


31 


30 


940842 


23733 


999983 


02 


940858 


23735 


059142 


30 


31 


7.955082 


22930 


9.999932 


02 


7.955100 


22981 


12.044900 


29 


32 


968870 


22273 


999981 


02 


■ 958389 


2-2275 


031111 


28 


33 


982233 


21608 


999980 


02 


932253 


21610 


017747 


27 


34 


995198 


20981 


999979 


02 


995219 


20983 


004781 


26 


35 


8.007787 


20390 


999977 


02 


8.007809 


20392 


11.992191 


25 


36 


020021 


19831 


999976 


02 


0-20045 


19333 


979955 


24 


37 


031919 


19302 


999975 


02 


031945 


19305 


958055 


23 


38 


043501 


18801 


999973 


02 


043527 


18803 


956473 


22 


39 


054781 


18325 


999972 


02 


054809 


18327 


945191 


21 


40 


065776 


17872 


999971 


02 


065806 


17874 


934194 


20 


41 


8.076500 


17441 


9.999969 


02 


8.076531 


17444 


11.923469 


19 


42 


086965 


17031 


999958 


02 


086997 


17034 


913003 


18 


43 


097183 


16639 


999956 


02 


097217 


16642 


902783 


17 


44 


107167 


16265 


999954 


03 


107202 


16-258 


892797 


16 


45 


116926 


15908 


999963 


03 


116953 


15910 


833037 


15 


46 


126471 


15566 


999951 


03 


126510 


15568 


873490 


14 


47 


135810 


15238 


999959 


03 


135351 


15241 


864149 


13 


48 


144953 


14924 


999958 


03 


144996 


149-27 


855004 


12 


49 


153907 


14622 


999956 


03 


153952 


146-27 


846048 


11 


50 


162681 


14333 


999954 


03 


162727 


14336 


837273 


10 


51 


8.171280 


14054 


9.999952 


03 


8.1713-28 


14057 


11.828672 


9 


52 


179713 


13786 


999950 


03 


179763 


13790 


8-20-237 


8 


53 


187985 


135-29 


999<)48 


03 


188036 


13532 


811964 


7 


54 


196102 


13-280 


999946 


03 


195156 


13-284 


803844 


6 


55 


204070 


13041 


999944 


03 


204126 


13044 


7i).>874 


5 


5() 


211895 


1-2810 


999942 


04 


211953 


1-2814 


788047 


4 


57 


219581 


1-2587 


999940 


04 


219541 


1-2590 


780359 


3 


58 


227134 


12372 


999938 


04 


227195 


12376 


77-2805 


2 


59 


234557 


12164 


999936 


04 


234621 


12Ui8 


765379 


1 


m 


241855 


11963 


999934 


04 


241921 


11967 


758079 





~1 


Cosine j 




Sine 1 


1 


Cotang. 




Tang. 


|M. 



89 Degrees. 







SINES 


AND TANGENTS. (1 Dc 


grec.) 




45 


M. 1 


Sine 1 


I>. 


Cosine 


D. 


Tang. 


D. 


Cotang. 







8.241855 


11963 


9.999934 


04 


8.241921 


11967 


11.758079 


60 


1 


249033 


11768 


999932 


04 


249102 


11772 


750898 


59 


2 


256094 


11580 


999929 


04 


256165 


11584 


743835 


58 


3 


263042 


11398 


999927 


04 


263115 


11402 


736885 


57 


4 


269881 


11221 


999925 


04 


269956 


11225 


730044 


56 


5 


276614 


11050 


999922 


04 


276C91 


11054 


723309 


55 


6 


283243 


10883 


999920 


04 


283323 


10887 


716677 


54 


7 


289773 


10721 


999918 


04 


289856 


10726 


710144 


53 


8 


296207 


10565 


999915 


04 


296292 


10570 


7037C8 


52 


9 


302546 


10413 


999913 


04 


302G34 


10418 


697366 


51 


10 


308794 


10266 


999910 


04 


308884 


10270 


691116 


50 


11 


8.314954 


10122 


9.999907 


04 


8.315046 


10126 


11.684954 


49 


12 


321027 


9982 


999905 


04 


321122 


9987 


678878 


48 


13 


327016 


9847 


999902 


04 


327114 


9851 


672886 


47 ■ 


14 


332924 


9714 


999899 


05 


333025 


9719 


666975 


46 , 


15 


338753 


9586 


999897 


05 


338856 


9590 


661144 


45 


16 


344504 


9460 


999894 


05 


344610 


9465 


655390 


44 


17 


350181 


9338 


999891 


05 


350289 


9343 


649711 


43 


18 


355783 


9219 


999888 


05 


355895 


9224 


644105 


42 


19 


361315 


9103 


999885 


05 


361430 


9108 


638570 


41 


20 


366777 


8990 


999882 


05 


366895 


8995 


633105 


40 


21 


8.372171 


8880 


9.999879 


05 


8.372292 


8885 


11.627708 


39 


22 


377499 


8772 


99J876 


05 


377622 


8777 


622378 


38 


23 


382762 


8667 


999873 


05 


282S89 


8672 


617111 


37 


24 


387962 


8564 


999870 


05 


388G92 


8570 


611908 


36 


25 


393101 


8464 


999867 


05 


393234 


8470 


606765 


35 


26 


398179 


8366 


999864 


05 


398315 


8371 


601685 


34 


27 


403199 


82T1 


999861 


05 


403338 


82'. 6 


5966G2 


33 


28 


408161 


8177 


999858 


05 


408304 


8182 


591696 


32 


29 


413068 


8088 


999854 


05 


413213 


8091 


586787 


31 


1 30 


417919 


7996 


999851 


06 


418068 


8002 


581932 


30 


21 


8.422717 


7909 


9.999848 


06 


8.422869 


7914 


11.577131 


29 


32 


427462 


7823 


999844 


06 


427618 


7830 


572382 


26 


33 


432156 


7740 


999841 


06 


432315 


7745 


567685 


27 


34 


436S00 


7657 


999838 


06 


436962 


76G3 


563038 


26 


35 


441394 


7577 


999834 


06 


441560 


7583 


558440 


25 


36 


445941 


7499 


999831 


06 


446110 


7505 


553890 


24 


37 


450440 


7422 


999827 


06 


450613 


7428 


549387 


23 


38 


454893 


7346 


999823 


06 


455070 


7352 


544930 


22 


39 


459301 


7273 


999820 


06 


459481 


7279 


540519 


21 


40 


463665 


7200 


999816 


06 


463849 


7206 


536151 


20 


41 


8.467985 


7129 


9.999812 


06 


8.468172 


7135 


11.531828 


19 


42 


472263 


7060 


999809 


06 


472454 


7066 


527546 


18 


43 


476498 


6991 


999805 


06 


476693 


6998 


523307 


17 


44 


480693 


6924 


999801 


06 


480892 


6931 


519108 


16 


45 


484848 


6859 


999797 


07 


485050 


6865 


514950 


15 


46 


4889C)3 


6794 


999793 


07 


489170 


6801 


510830 


14 


47 


493040 


6731 


999790 


07 


493250 


6738 


506750 


13 


48 


497078 


6669 


999786 


07 


497293 


6676 


502707 


12 


49 


501080 


6608 


999782 


07 


501298 


6615 


498702 


11 


50 


505045 


6548 


999778 


07 


505267 


6555 


494733 


10 


51 


8.508974 


6489 


9.999774 


07 


8.509200 


6496 


11.490800 


9 


52 


5128G7 


6431 


999769 


07 


513098 


6439 


486902 


8 


53 


516726 


6375 


999765 


07 


516961 


6382 


483039 


7 


54 


520551 


6319 


999761 


07 


520790 


6326 


479210 


6 


55 


524343 


6264 


999757 


07 


524586 


6272 


475414 


5 


56 


528102 


6211 


999753 


07 


528349 


6218 


471651 


4 


57 


531828 


6158 


999748 


07 


532080 


6165 


467920 


3 


58 


535523 


6106 


999744 


07 


535779 


6113 


464221 


2 


59 


539186 


6055 


999740 


07 


539447 


6062 


460553 


1 


60 


542819 


6004 


999735 


07 


543084 


6012 


456916 







Cosine 


1 


Sine 




Co tang. 


1 


1 Tang. 


IM. 



88 Degrees. 



46 




(2 Degrees.) a 


TABLE OF LOGARITHMIC 




1 M. 

1 


Sine 


D. 


Cosiae. 


D. 


Tang 


D 


Cotang. 


J 





8.542819 


6004 


0.999735 


07 


8.. 543084 


6012 


11.458916 


60 


1 


54()422 


5955 


999731 


07 


540631 


5982 


453309 


59 


2 


.54')935 


5936 


939726 


07 


550268 


5914 


443732 


58 


1 3 


553539 


5858 


999722 


08 


553817 


5866 


44!)183 


57 


4 


557054 


5811 


993717 


08 


5573:6 


5319 


442664 


56 


5 


5)0540 


5705 


993713 


08 


5608-28 


5773 


439172 


55 


6 


503939 


5719 


993708 


08 


564291 


5727 


435709 


54 


7 


567431 


5074 


939704 


08 


557727 


5682 


432273 


53 


8 


57>)8 >0 


5033 


999699 


08 


571137 


5538 


428863 


52 


9 


5742H 


5587 


993694 


03 


574520 


5595 


425480 


51 


10 


577566 


5544 


993089 


08 


577877 


5552 


422123 


50 


11 


8.58 892 


5502 


9.939685 


C8 


8.581208 


5510 


11.418792 


49 


12 


5841J3 


5460 


939383 


08 


534514 


5168 


415486 


48 


13 


587469 


5119 


999375 


08 


587795 


5127 


412205 


47 


14 


593721 


5379 


939370 


08 


591051 


5387 


438949 


46 


15 


593948 


5339 


9936G5 


08 


534233 


5347 


405717 


45 


16 


597152 


53 JO 


999860 


08 


597492 


5338 


432508 


41 


17 


0J0332 


5201 


939355 


08 


G00G77 


5270 


399323 


43 


.18 


C03483 


5223 


933550 


08 


603839 


5232 


.396161 


42 


1 19 


G06523 


5136 


939345 


09 


605978 


5194 


333022 


n 


20 


G09734 


5149 


993340 


09 


G10094 


5158 


389306 


40 


21 


8.612823 


5112 


9.993335 


09 


8.613189 


5121 


11.383811 


39 


22 


015891 


5J73 


993329 


03 


616262 


5085 


383738 


38 


23 


C18337 


5341 


993624 


03 


619313 


5350 


380687 


37 


24 


C21932 


539) 


999619 


09 


622343 


5015 


377657 


36 


25 


021935 


4972 


993314 


09 


6-25352 


4931 


374648 


35 


26 


627918 


4933 


9J3608 


09 


628340 


4947 


371660 


34 


27 


630911 


4934 


939803 


03 


631308 


4913 


368692 


33 


28 


033854 


4871 


939597 


09 


634256 


4883 


335744 


32 


29 


636776 


4339 


939592 


OJ 


637184 


4848 


362316 


31 


30 


639383 


4836 


9335tt6 


09 


643093 


4816 


353937 


33 


31 


8.642583 


4775 


9.933581 


09 


8.642982 


4784 


11.357013 


29 


32 


645428 


47 3 


993575 


09 


645853 


4753 


354147 


28 


33 


648274 


4712 


933570 


09 


648704 


4722 


351-296 


27 


34 


651102 


4682 


933564 


09 


651537 


4331 


343463 


23 


35 


653911 


4652 


933558 


10 


654352 


4381 


345648 


25 


36 


656702 


4622 


993->53 


10 


657149 


4631 


342851 


24 


37 


659475 


4592 


999547 


10 


659923 


4602 


340072 


23 


38 


602-233 


4533 


993541 


13 


632689 


4573 


337311 


22 


39 


664968 


4535 


999535 


10 


665433 


4514 


334567 


21 


40 


667689 


4536 


939529 


10 


668 J 60 


4523 


331340 


20 


41 


8.670393 


4479 


9.999524 


10 


8.670870 


4488 


11.329130 


19 


42 


673083 


4451 


993518 


10 


673563 


4431 


326437 


18 


43 


675751 


4424 


999512 


10 


676239 


4434 


323761 


17 


44 


678405 


4397 


999536 


13 


678903 


4417 


321100 


16 


45 


681043 


4370 


993500 


10 


681544 


4333 


318456 


15 


46 


683665 


4344 


933493 


10 


684172 


4354 


315828 


14 


47 


686272 


4318 


939487 


13 


686784 


4323 


313216 


13 


48 


688863 


42J2 


999481 


10 


689381 


4303 


310519 


12 


49 


691438 


4267 


939475 


10 


691963 


4277 


308037 


11 


50 


693998 


4242 


999469 


10 


694529 


4252 


305471 


10 


51 


8.096543 


4217 


9.999463 


11 


8.697081 


4228 


11.302919 


9 


52 


699073 


4192 


993455 


11 


699317 


4203 


303383 


8 


53 


701.589 


4168 


999450 


11 


702139 


4179 


297861 


7 


54 


704090 


4144 


939443 


11 


704646 


4155 


295354 


6 


55 


706577 


4121 


' 939437 


11 


707140 


4132 


292860 


5 


56 


709049 


4037 


999431 


11 


709618 


4108 


293382 


4 


57 


711.507 


4074 


999124 


11 


712083 


4085 


237917 


3 


58 


713952 


4051 


999418 


11 


714534 


4062 


'28.5465 


2 


59 


716:i83 


4029 


999411 


n 


716972 


4040 


28:1028 


1 


60 


718800 


4006 


999404 


11 


719398 


4317 


280604 








Cosine 




Sine 




Cotaug. 


1 


Tang. 


1 M 



87 Degreea 



SINES AND TANGENTS. (3 Degrees.) 



47 



iM. 


1 Sine 


1 D- 


1 Cosine 


1 D. 


1 Tang. 


1 D. 


1 Cotang. 


1 





8.718800 


4006 


9.999404 


11 


8.719396 


4017 


11.280604 


60 


1 


721204 


3984 


999398 


11 


721806 


3995 


278194 


59 


2 


723595 


3932 


999391 


11 


724204 


3974 


275796 


58 


3 


725972 


3941 


999384 


11 


723588 


3952 


273412 


57 


4 


728337 


3919 


999378 


11 


728959 


3930 


271041 


56 


5 


730688 


3898 


999371 


11 


731317 


3909 


268683 


55 


6 


733027 


3877 


999304 


12 


733663 


3889 


266337 


54 


7 


735354 


3857 


999357 


12 


735993 


3808 


264004 


53 


8 


737067 


3836 


909350 


12 


738317 


3848 


261683 


52 


9 


739969 


3816 


999343 


12 


740026 


3827 


259374 


51 


10 


742259 


3793 


999336 


U 


742922 


3807 


257078 


50 


11 


8.744536 


3776 


9.999329 


12 


8.745207 


3787 


11.254793 


49 


12 


746802 


3756 


99J322 


12 


747479 


3768 


252521 


48 


13 


749055 


3737 


999315 


12 


749740 


3749 


250260 


47 


14 


751297 


3717 


999308 


12 


751989 


3729 


2«8011 


46 


15 


753528 


3698 


999301 


12 


754227 


3710 


245773 


45 


16 


755747 


3679 


999294 


12 


756453 


3392 


243547 


44 


]7 


757955 


3661 


999236 


12 


758668 


3673 


241332 


43 


]8 


760151 


3642 


999279 


12 


760872 


3655 


239128 


42 


19 


762337 


3624 


999272 


12 


763065 


3636 


236935 


41 


20 


704511 


360G 


999265 


12 


765246 


3618 


234754 


40 


21 


8.706675 


3588 


9.999257 


12 


8.767417 


3600 


11.232583 


39 


22 


768828 


3570 


999250 


13 


709578 


3583 


230422 


38 


23 


770970 


3553 


999242 


13 


771727 


3565 


228273 


37 


24 


773101 


3535 


999235 


13 


773866 


3548 


226134 


36 


25 


775223 


3518 


999227 


13 


775995 


3531 


224005 


35 


26 


777333 


35J1 


999220 


13 


778114 


3514 


221886 


34 


27 


779434 


3484 


999212 


13 


780222 


3497 


219778 


33 


28 


781524 


3467 


999205 


13 


782320 


3480 


217680 


32 


29 


783605 


3451 


999197 


13 


784408 


3464 


215592 


31 


30 


785675 


3431 


999189 


13 


786486 


3447 


213514 


30 


31 


8.787736 


3418 


9.999181 


13 


8.788554 


3431 


11.211446 


29 


32 


789787 


3402 


999174 


13 


790613 


3414 


209387 


28 


33 


791828 


3386 


999166 


13 


792662 


3399 


207338 


27 


34 


793859 


3370 


999158 


13 


794701 


3383 


205299 


26 


35 


795881 


3354 


999150 


13 


796731 


3368 


203269 


25 


36 


797894 


3339 


999142 


13 


798752 


3352 


201248 


24 


37 


799897 


3323 


999134 


13 


800763 


3337 


199237 


23 


38 


801892 


3308 


999123 


13 


802765 


3322 


197235 


22 


1 39 


803876 


3293 


999118 


13 


804758 


3307 


195242 


21 


i 40 


805852 


3278 


999110 


13 


808742 


3292 


193258 


20 


i 41 


8.8G7819 


3263 


9.999102 


13 


8.808717 


3278 


11.191283 


19 


42 


809777 


3249 


999094 


14 


810(>83 


3262 


189317 


18 


43 


811723 


3234 


999086 


14 


812641 


3248 


187359 


17 


44 


813867 


3219 


999077 


14 


814589 


3233 


185411 


16 


45 


815599 


3205 


999069 


14 


816529 


3219 


183471 


15 


46 


817522 


3191 


999061 


14 


818461 


3205 


181539 


14 


47 


819436 


3177 


999053 


14 


820384 


3191 


179616 


13 


48 


821343 


3163 


999044 


14 


822298 


3177 


177702 


12 


49 


823240 


3149 


999036 


14 


824205 


3163 


175795 


11 


50 


825130 


3135 


999027 


14 


826103 


3150 


173897 


10 


51 


8.827011 


3122 


9.999019 


14 


8.827992 


3136 


11.172008 


9 


52 


828884 


3108 


999010 


14 


829874 


3123 


170126 


8 


53 


830749 


3095 


999002 


14 


831748 


3110 


168252 


7 


54 


832607 


3082 


998993 


14 


833613 


3096 


166387 


6 


55 


834456 


'3069 


998934 


14 


835471 


3083 


164529 


5 


56 


836297 


3056 


998976 


14 


837321 


3070 


162679 


4 


57 


838130 


3043 


998967 


15 


839163 


3057 


160837 


3 


58 


839956 


3030 


998958 


15 


840998 


3045 


159002 


2 


59 


841774 


3017 


9J8950 


15 


842825 


3032 


157175 


1 


60 


843585 


3000 


998911 


15 


844644 


3019 


155356 





1 


Cosine 1 




Sine 


1 


Co tang. 1 


1 


Tang. 1 


M. 



86 I>egrees 



48 




(4 Degrees.) a 


TABLE OF LOGARITHMIC 






M. 1 


Sino 


D. 1 


Cosine 


D. 


Tang. 


D 


Cotang i 









8.843585 


3005 


9.998941 


15 


8.844644 


3019 


11.155 $56 


60 




1 


815387 


2992 


998932 


15 


846455 


3007 


153545 


59 




2 


847183 


2980 


098923 


15 


848260 


2995 


151740 


58 




3 


848971 


2967 


998914 


15 


850057 


2932 


149943 


57 




4 


850751 


2955 


998905 


15 


851846 


2970 


148154 


56 




5 


852525 


2943 


998896 


15 


853628 


2958 


146372 


55 




6 


8542Ji 


2931 


998387 


15 


85.5403 


2946 


144597 


54 




7 


85604'J 


2919 


998873 


15 


8.57171 


293.5 


142829 


53 




8 


8573J1 


2937 


998869 


15 


858932 


2923 


141008 


52 




9 


859546 


2896 


998860 


15 


860686 


2911 


139314 


51 




10 


861283 


2884 


993851 


15 


882433 


2900 


137567 


50 




11 


8.863014 


2873 


9.998841 


15 


8.864173 


2888 


11.135827 


49 




12 


864733 


21^61 


998332 


15 


815906 


2877 


134094 


48 




13 


866455 


2350 


993323 


1(5 


867632 


2866 


132368 


47 




14 


838165 


2839 


993313 


16 


869351 


2854 


130649 


46 




15 


869308 


2828 


993804 


16 


871064 


2843 


128936 


45 




16 


871565 


2817 


993795 


16 


872770 


2832 


127230 


44 




17 


873255 


2306 


998785 


16 


874469 


2821 


125531 


43 




18 


874938 


2795 


993776 


16 


876162 


2811 


123838 


42 




Id 


876615 


2786 


993766 


16 


877849 


280 J 


122151 


41 




20 


878285 


2773 


993757 


16 


879529 


2739 


120471 


40 




21 


8.879949 


2763 


9.998747 


16 


8.881202 


2779 


11.118793 


39 




22 


881607 


2752 


993738 


16 


83-2869 


27(58 


117131 


38 




23 


883258 


2742 


993728 


16 


884530 


2758 


115470 


37 




24 


88490-3 


2731 


998718 


16 


886185 


2747 


113815 


36 




25 


8815542 


2721 


998708 


13 


887833 


2737 


112167 


35 




26 


888174 


2711 


998')99 


16 


889476 


2727 


110524 


34 




27 


889801 


2700 


9J8G89 


16 


891112 


2717 


108888 


33 




28 


891421 


2690 


993679 


16 


892742 


2707 


107258 


32 




2D 


893035 


2680 


998669 


17 


894366 


2697 


105634 


31 




30 


894643 


2670 


998659 


17 


895984 


2087 


101016 


30 




31 


8.896246 


2660 


9.993649 


17 


8.897598 


2677 


11.102404 


29 




32 


897842 


2051 


993639 


17 


899203 


2667 


100797 


26 




33 


899432 


2641 


993)29 


17 


900803 


2658 


099197 


27 




34 


901017 


3631 


993619 


17 


902398 


2643 


097602 


26 




35 


902598 


2622 


998609 


17 


903987 


2638 


096013 


25 




36 


904169 


2612 


998599 


17 


905570 


2629 


094430 


24 




37 


905736 


2603 


998539 


17 


907147 


2620 


092353 


23 




38 


907297 


2593 


993578 


17 


908719 


2610 


091281 


22 




39 


908853 


2584 


998568 


17 


910235 


2601 


089715 


21 




40 


910404 


2575 


998558 


17 


911840 


2592 


088154 


20 




41 


8.911949 


2566 


9.998548 


17 


8.913401 


2583 


11.086599 


19 




42 


913488 


2556 


998537 


17 


914951 


2574 


085049 


18 




43 


915022 


2547 


998527 


17 


916495 


2565 


oe:»05 


17 




44 


916550 


2538 


998516 


18 


918034 


2556 


081966 


16 




45 


918073 


2529 


998506 


18 


919563 


2547 


080432 


15 




46 


919591 


2520 


998495 


18 


921096 


2538 


078904 


14 




47 


921103 


2512 


993485 


m 


922619 


2530 


077381 


13 




48 


922610 


2503 


998474 


18 


924136 


2521 


07.58(54 


12 




49 


924112 


2494 


998164 


18 


925649 


2512 


074351 


11 




50 


925009 


2486 


993453 


18 


927156 


2503 


07-2344 


10 




51 


8.927100 


2477 


9.998442 


18 


8.928658 


2495 


11.071342 


9 




52 


928587 


2469 


998431 


18 


930155 


2486 


069345 


8 




53 


9300()8 


2460 


998421 


18 


931647 


2478 


0(58353 


7 




54 


931544 


2452 


998410 


18 


933134 


2470 


0()(58()6 


6 




55 


933015 


2443 


993399 


18 


934616 


2461 


0(55:J84 


5 




56 


934481 


2435 


993333 


18 


936093 


2453 


063907 


4 




57 


935942 


2427 


993377 


18 


9375()5 


2445 


062435 


3 




58 


937398 


2419 


9933()6 


18 


939032 


2437 


0()0968 


2 




59 


938850 


2411 


998355 


18 


940494 


2430 


05950(5 


1 




m 


940296 


2403 


998344 


18 


941952 


2421 


058048 









1 Cosine 


1 


1 Siue 


1 


Cotang. 


1 


1 Tang. 


1 M 





85 Degrees. 







SINES 


AND TANGENTS. (5 DegrCCS.) 




49 


M. 


1 Sine 


1 ». 


1 Cosine | D. 


1 Tang. 


1 I>. 


1 Cotang. 


1 







8.940296 


2403 


9.998344 


19 


8.941952 


2421 


11.058048 


60 




1 


941738 


2394 


998333 


19 


943404 


2413 


056596 


59 




2 


943174 


2387 


9983-22 


19 


944852 


2405 


055148 


58 




3 


944606 


2379 


998311 


19 


946295 


2397 


053705 


57 




4 


946034 


2371 


998300 


19 


947734 


2390 


052266 


56 




5 


947456 


2363 


998289 


19 


949168 


2382 


050832 


55 




6 


948874 


2355 


998277 


19 


950597 


2374 


049403 


54 




7 


950287 


2348 


998266 


19 


952021 


2366 


047979 


53 




8 


951698 


2340 


998255 


19 


953441 


2360 


046559 


52 




9 


953100 


2332 


998243 


19 


954856 


2351 


045144 


51 




10 


954499 


2325 


998232 


19 


956267 


2344 


043733 


50 




11 


8.955894 


2317 


9.998220 


19 


8.957674 


2337 


11.042326 


49 




12 


957284 


2310 


998209 


19 


959075 


2329 


040925 


48 




13 


958670 


23D2 


998197 


19 


980473 


2323 


039527 


47 




14 


960052 


2295 


998186 


19 


981886 


2314 


038134 


46 




15 


961429 


2288 


998174 


19 


983255 


2307 


036745 


45 




16 


962801 


2280 


998163 


19 


964639 


2300 


035361 


44 




17 


964170 


2273 


998151 


19 


966019 


2293 


033981 


43 




18 


965534 


2266 


998139 


20 


967394 


2286 


032606 


42 




19 


966893 


2259 


998128 


20 


968766 


2279 


031234 


41 




20 


968249 


2252 


998116 


20 


970133 


2271 


029867 


40 




21 


8.969600 


2244 


9.998104 


20 


8.971496 


2265 


11.028504 


39 




22 


970947 


2238 


998092 


20 


972855 


2257 


027145 


38 




23 


972289 


2231 


998080 


20 


974209 


2251 


025791 


37 




24 


973628 


2224 


998068 


20 


975560 


2244 


024440 


36 




25 


974962 


2217 


998056 


20 


976906 


2237 


023094 


35 




26 


976293 


2210 


998044 


20 


978248 


2230 


021752 


34 




27 


977619 


2203 


998032 


20 


979586 


2223 


020414 


33 




28 


978941 


2197 


998020 


20 


930921 


2217 


019079 


32 




29 


980259 


2190 


998008 


20 


932251 


2210 


017749 


31 




30 


981573 


2183 


997998 


20 


983577 


2204 


016423 


30 




31 


8.982883 


2177 


9.997934 


20 


8.984899 


2197 


11.015101 


29 




32 


984189 


2170 


997972 


20 


985217 


2191 


013783 


28 




33 


985*91 


2163 


997959 


20 


987532 


2184 


012468 


27 




34 


986789 


2157 


997947 


20 


988842 


2178 


011158 


26 




35 


988083 


2150 


997935 


21 


990149 


2171 


009851 


25 




36 


989374 


2144 


997922 


21 


991451 


2165 


008549 


24 




37 


990660 


2138 


997910 


21 


992750 


2158 


097250 


23 




38 


991943 


2131 


937897 


21 


994045 


2152 


005955 


22 




39 


993222 


2125 


997885 


21 


995337 


2146 


004863 


21 




40 


994497 


2119 


997872 


21 


995624 


2140 


003376 


20 




41 


8.995768 


2112 


9.997830 


21 


8.997903 


2134 


11.002092 


19 




42 


997036 


2106 


997847 


21 


999188 


2127 


000812 


18 




43 


998299 


2100 


997835 


21 


9.000465 


2121 


10.999535 


17 




44 


999560 


2094 


907822 


21 


001733 


2115 


993262 


16 




45 


9.000816 


2087 


997809 


21 


003007 


2109 


998993 


15 




46 


002069 


2082 


997797 


21 


004272 


2103 


995728 


14 




47 


003318 


2076 


997784 


21 


005534 


2097 


994466 


13 




48 


004563 


2070 


997771 


21 


006792 


2091 


993208 


12 




49 


005805 


2034 


997758 


21 


008047 


2085 


991953 


11 




50 


007044 


2058 


997745 


21 


009298 


2080 


990702 


10 




51 


9.008278 


2052 


9.997732 


21 


9.010543 


2074 


10.989454 


9 




52 


009510 


2046 


997719 


21 


011790 


2068 


938210 


8 




53 


0J0737 


2040 


997706 


21 


013031 


2062 


986959 


7 




54 


011962 


2034 


997693 


22 


014268 


2056 


985732 


6 




55 


013182 


2029 


997680 


22 


015502 


2051 


984498 


5 




56 


014400 


2023 


997667 


22 


016732 


2045 


983268 


4 




57 


015613 


2017 


997654 


22 


017959 


2040 


982041 


3 




58 


016824 


2012 


997641 


22 


019183 


2033 


980817 


2 




59 


018031 


2006 


997628 


22 


020403 


2028 


979597 


1 




60 


019235 


2000 


997614 


22 


021620 


2023 


978380 


o| 






1 Cosine 




Sine 1 


Cotang. 1 


1 


Tang. 1 


M. 










84 Degn 


jes. 












ELLWO< 


3d's TE 


ST PROB. 


/ 


L 











60 




(6 Degrees.) a 


TABLE OF LOGARITHMIC 




M. 


1 Sine 


1 D. 


1 Cosine 


! D 

1 2r2 


1 Tang. 


! D. 


1 Cotang. 


'^ 





9.019235 


1 2000 


9.997614 


9.021620 


2023 


10.978380 


1 60 


1 


020435 


1995 


997601 1 22 


0228.34 


2017 


977166 


59 1 


2 


021632 


1989 


997588 i 22 


024044 


2011 


975956 


58 


3 


022825 


1984 


997574 


22 


0252,51 


20n6 


974749 


1 57 i 


4 


024016 


1978 


997561 


22 


026455 


2UO0 


973545 


i 56 


5 


025203 


1973 


997547 


22 


027655 


1995 


972345 


55 


6 


020386 


1907 


997534 


23 


028852 


1990 


971148 


54 


7 


027567 


1962 


997520 


23 


030046 


1985 


969954 


53 


8 


028744 


1957 


997507 


23 


031237 


1979 


968763 


52 


9 


029918 


1951 


997493 


23 


032425 


1974 


967575 


51 


10 


031089 


1947 


997480 


23 


033C09 


1969 


966391 


50 


11 


9.032257 


1941 


9.997466 


23 


9.034791 


1964 


10.965209 


49 


12 


033421 


1936 


997452 


23 


035969 


1958 


9(>4031 


48 


13 


034582 


1930 


! 997439 


23 


037144 


1953 


962856 


47 


14 


035741 


1925 


997425 


23 


038316 


1948 


901684 


46 


15 


036896 


1920 


997411 


23 


039485 


1943 


960515 


45 


16 


038048 


1915 


997397 


23 


040651 


19:^8 


959349 


44 


17 


039197 


1910 


997383 


23 


041813 


1933 


958187 


43 


18 


040342 


1905 


997369 


23 


042973 


1928 


957027 


4-2 


19 


041485 


1899 


997355 


23 


044130 


1923 


955870 


41 


20 


042625 


1894 


997341 


23 


045284 


1918 


954716 


40 


21 


9.043762 


1889 


9.997327 


24 


9.046434 


1913 


10.953.566 


39 


22 


044895 


1884 


997313 


24 


047582 


1908 


952418 


38 


23 


046026 


1879 


997299 


24 


048727 


1903 


951273 


37 


24 


047154 


1875 


997285 


24 


049869 


1898 


950131 


36 


25 


048279 


1870 


997271 


24 


051008 


1893 


948992 


35 


26 


049400 


1865 


997257 


24 


052144 


1889 


947856 


34 


27 


050519 


1860 


997242 


24 


053277 


1884 


946723 


33 


28 


051635 


1855 


997228 


24 


054407 


1879 


945593 


32 


29 


052749 


1850 


997214 


24 


055535 


1874 


944465 


31 


30 


053859 


1845 


997199 


24 


056659 


1870 


943341 


30 


31 


9.054966 


1841 


9.997185 


24 


9.0.57781 


1805 


10.942219 


29 


32 


056071 


1836 


997170 


24 


0589D0 


1869 


941100 


28 


33 


057172 


1831 


997156 


24 


060016 


1855 


939984 


27 


34 


0.58271 


1827 


997141 


24 


061130 


1851 


938870 


26 


35 


059367 


1822 


997127 


24 


062240 


1846 


937760 


25 


36 


060460 


1817 


997112 


24 


063348 


1842 


936652 


24 


37 


061551 


1813 


997098 


24 


064453 


1837 


935547 


23 


38 


062639 


1808 


997083 


25 


065556 


1833 


034444 


22 


39 


063724 


1804 


9970G8 


25 


066655 


1828 


933345 


21 


40 


064806 


1799 


997053 


25 


067752 


1824 


932248 


20 


41 


9.005885 


1794 


9.997039 


25 


9.068846 


1819 


10.931154 


19 


42 


06691)2 


1790 


997024 


25 


069938 


1815 


930062 


18 


43 


0G8036 


1786 


997009 


25 


071027 


1810 


928973 


17 


44 


069107 


1781 


993994 


25 


072113 


1806 


927887 


16 


45 


070176 


1/// 


996979 


25 


073197 


1802 


926803 


15 


46 


071242 


1772 


996964 


25 


074278 


1797 


925722 


14 


47 


072306 


17C>8 


996949 


25 


075356 


1793 


924644 


13 


48 


073306 


1763 


996934 


25 


076432 


1789 


923508 


12 


49 


074424 


1759 


99r>919 


25 


077505 


1784 


922495 


11 


50 


075480 


1755 


996904 


25 


078576 


1780 


921424 


10 


51 


9.076533 


1750 


9.993889 


25 


9.079044 


1776 


10.920356 


9 


52 


077583 


1746 


990874 


25 


080710 


1772 


919290 


8 


53 


078631 


1742 


996858 


25 


081773 


1767 


918227 


7 


54 


079676 


1738 


996843 


25 


082833 


1763 


917167 


6 


55 


080719 


1733 


990828 


25 


08.3891 


1759 


916109 


5 


56 


081759 


1729 


996812 


26 


a'^4;)47 


1755 


9150.53 


4 


57 


082797 


1725 


990797 


26 


08(5000 


1751 


914000 


3 


58 


083832 


1721 


99<)782 


26 


087050 


1747 


9129.50 


o 


59 


0848()4 


1717 


9967()6 


26 


088098 


1743 


911902 


1 


GO 


085894 


1713 


996751 


26 


089144 


17:J8 


91085() 





1 


Cosine | 


1 


Sine 1 1 


Cotang. 1 


1 


Tang. 1 


-M.| 



83 Degrees 







SINES 


AND TANGENTS. (7 Degrees.) 




51 


M. 


1 Sine 


i D- 


1 Cosine 


D. 


1 Tang. 


1 D- 


Cotang. 







9.085894 


1713 


9.996751 


26 


9.089144 


1738 


10.9108.56 


60 


1 


086922 


1709 


996735 


26 


090187 


1734 


909813 


59 


2 


087947 


1704 


99r)720 


26 


091228 


1730 


908772 


58 


3 


088970 


1700 


996704 


26 


092266 


1727 


907734 


57 


4 


089990 


1696 


996688 


26 


093302 


1722 


906698 


56 


5 


091008 


1692 


996673 


26 


094336 


1719 


905664 


55 


6 


092024 


1688 


9966.57 


26 


095367 


1715 


904633 


54 


7 


093037 


1684 


996641 


28 


096395 


1711 


903605 


53 


8 


094047 


1680 


996625 


26 


097422 


1707 


902578 


52 


9 


095056 


1676 


995610 


26 


098446 


1703 


901.554 


51 


10 


096062 


1673 


996594 


26 


099468 


1699 


9005.32 


50 


11 


9.097065 


1668 


9.996578 


27 


9.100487 


1695 


10.899513 


49 


12 


098066 


1665 


996562 


27 


101504 


1691 


898496 


48 


13 


099065 


1661 


996546 


27 


102519 


1687 


897481 


47 


14 


100062 


1657 


996530 


27 


103532 


1684 


898468 


46 


15 


101056 


1653 


996514 


27 


104542 


1680 


895458 


45 


16 


102048 


1649 


996498 


27 


105550 


1676 


894450 


44 


17 


103037 


1645 


996482 


27 


106556 


1672 


8!)3444 


43 


18 


104025 


1641 


996465 


27 


107.559 


1669 


892441 


42 


19 


105010 


1638 


996449 


27 


103560 


1685 


891440 


41 


20 


105992 


1634 


996433 


27 


109559 


1661 


890441 


40 


21 


9.106973 


1630 


9.996417 


27 


9.1105.56 


1658 


10.889444 


39 


22 


107951 


1627 


996400 


27 


111551 


1854 


888449 


38 


23 


108927 


1623 


996384 


27 


112543 


1650 


887457 


37 


24 


109901 


1619 


996368 


27 


113533 


1646 


888467 


36 


25 


110873 


1616 


99G351 


27 


114521 


1843 


885479 


35 


28 


111842 


1612 


996335 


27 


115507 


1639 


884493 


34 


27 


112809 


1608 


996318 


27 


116491 


1636 


883509 


33 


28 


113774 


1605 


996302 


28 


117472 


1832 


882528 


32 


29 


114737 


1601 


996285 


28 


118452 


1829 


831548 


31 


30 


115698 


1597 


996269 


28 


119429 


1625 


880571 


30 


31 


9.116656 


1594 


9.996252 


28 


9.120404 


1622 


10.879596 


29 


32 


117613 


1590 


996235 


28 


121377 


1618 


878623 


28 


33 


118567 


1587 


996219 


28 


122348 


1615 


877652 


27 


34 


119519 


1583 


99G202 


28 


123317 


1611 


876683 


26 


35 


120469 


1580 


996185 


28 


124284 


1607 


87.5716 


25 


36 


121417 


1576 


996168 


28 


125249 


1604 


874751 


24 


37 


122362 


1573 


996151 


28 


126211 


1801 


873789 


23 


38 


123306 


1569 


996134 


28 


127172 


1597 


872828 


22 


39 


124248 


1566 


996117 


28 


128130 


1594 


871870 


21 


40 


125187 


1562 


996100 


28 


129087 


1591 


870913 


20 


41 


9.126125 


1559 


9.996083 


29 


9.130041 


1587 


10.869959 


19 


42 


127060 


1556 


996066 


29 


130994 


1584 


889006 


18 


43 


127993 


1552 


996049 


29 


131944 


1581 


888056 


17 


44 


128925 


1549 


996032 


29 


132893 


1577 


867107 


16 


45 


129854 


1545 


996015 


29 


133839 


1574 


866161 


15 


46 


130781 


1542 


995998 


29 


134784 


1571 


86.5216 


14 


47 


131706 


1539 


995980 


29 


135726 


1567 


864274 


13 


48 


132630 


1535 


995963 


29 


136667 


1584 


863333 


12 


49 


133551 


1532 


995946 


29 


137605 


1581 


862395 


11 


50 


134470 


1529 


995928 


29 


138542 


1558 


861458 


10 


51 


9.135387 


1525 


9.995911 


29 


9.139476 


1555 


10.860524 


9 


52 


136303 


1522 


995894 


29 


140409 


1551 


859591 


8 


53 


137216 


1519 


995876 


29 


141340 


1548 


858660 


7 


54 


138128 


1515 


995859 


29 


142269 


1545 


857731 


6 


55 


139037 


1512 


995841 


29 


143196 


1542 


856804 


5 


56 


139944 


1509 


995823 


29 


144121 


1539 


855879 


4 


57 


140850 


1506 


995806 


29 


145044 


1535 


854956 


3 


58 


141754 


1503 


995788 


29 


145966 


1532 


854034 


2 


59 


142555 


1500 


995771 


29 


146885 


1529 


853115 


1 


60 


143555 


1496 


995753 


29 


147803 


1526 


852197 





Cosine 




I Sine 




Cotang. 




1 Tang. 


M. 



82 Degrees. 



52 



(8 Degrees.) a table of logarithmic 



|M. 


1 Sine 


D 


1 Cosine 


D. 


1 Tang. 


1 D 


1 Cotang. 




1 


9.143555 


1496 


9.9957.53 


30 


9.147803 


1526 


10.852197 


60 1 


1 1 


144453 


1493 


995735 


30 


148718 


1523 


851282 


59 1 


2 


145349 


1490 


995717 


30 


149^)32 


1520 


850368 


58 


3 


146243 


1487 


995699 


30 


150544 


1517 


849456 


57 


4 


147136 


1484 


995681 


30 


151454 


1514 


848546 


56 


5 


148026 


1481 


995664 


30 


152363 


1511 


847637 


55 


6 


1489 J 5 


1478 


995646 


30 


1532'-9 


1508 


846731 


54 


7 


149802 


1475 


995(-28 


30 


154174 


1505 


845826 


53 


8 


150G86 


1472 


995610 


30 


155077 


1502 


844923 


52 


9 


15J5()9 


i4n9 


995591 


30 


155978 


1499 


844022 


51 


10 


152451 


1460 


995573 


30 


156877 


1496 


843123 


50 


11 


9.153330 


1463 


9 995555 


30 


9.1.)7775 


1493 


10.842225 


49 


12 


1542*8 


14!;0 


995537 


30 


158671 


1490 


841329 


4b 


13 


155083 


1457 


995519 


30 


159565 


1487 


840435 


47 


14 


155957 


14.54 


995501 


31 


160457 


1484 


839543 


46 


15 


150830 


1451 


995482 


31 


161347 


14«1 


838C53 


45 


16 


1.57700 


1148 


995464 


31 


162236 


1479 


837764 


44! 


17 


1585fi9 


1445 


995446 


31 


163123 


1476 


836877 


43! 


18 


159435 


1442 


995427 


31 


164008 


1473 


835992 


42 


19 


160301 


1439 


995409 


31 


164"S92 


1470 


835108 


41 


20 


161164 


1436 


995390 


31 


165774 


1467 


834226 


40 


21 


9.162025 


1433 


9.995372 


31 


9.1666.54 


1464 


10.833346 


39 


22 


162885 


1430 


995353 


31 


167532 


1461 


832468 


38 


23 


163743 


1427 


995334 


31 


168409 


14.58 


831591 


37 


24 


164600 


1424 


99.5316 


31 


169284 


1455 


830716 


36 


25 


165454 


1422 


995-297 


31 


170157 


1453 


829S43 


35 


26 


16G307 


1419 


995278 


31 


171029 


1450 


828971 


34 


27 


167159 


1416 


995200 


31 


171899 


1447 


828101 


33 i 


28 


168008 


1413 


995241 


32 


172767 


1444 


827233 


32 


29 


168856 


1410 


995222 


32 


173634 


1442 


8263C6 


31 


30 


169702 


1407 


995203 


32 


174499 


1439 


825501 


30 


31 


9.170547 


1405 


9.995184 


32 


9.175362 


14.36 


10.824638 


29 


32 


171389 


1402 


9951t;5 


32 


176224 


1433 


823776 


28 ^ 


33 


172230 


1399 


995146 


32 


177084 


1431 


822916 


27 


34 


. 173070 


1396 


995127 


32 


177942 


1428 


822058 


26 


35 


173908 


1394 


995108 


32 


178799 


1425 


821201 


25 


36 


174744 


1391 


995089 


32 


179655 


1423 


820345 


24 


37 


175578 


1388 


995070 


32 


1805C8 


1420 


819492 


23 


38 


176411 


1386 


995051 


32 


181360 


1417 


818G40 


22 


39 


177242 


1383 


995032 


32 


182211 


1415 


817789 


21 


40 


178072 


1380 


995013 


32 


183059 


1412 


816941 


20 


41 


9.178900 


1377 


9.994993 


32 


9.183907 


1409 


10.816093 


19 


42 


179726 


1374 


994974 


32 


184752 


1407 


815248 


18 


43 


180551 


1372 


994955 


32 


185597 


1404 


814403 


17 


44 


181374 


1369 


994935 


32 


186439 


1402 


813561 


16 


45 


18-2196 


1366 


994916 


33 


187280 


1399 


812720 


15 


46 


183016 


1364 


994896 


33 


188120 


1396 


811880 


14 


47 


183834 


1361 


994877 


33 


188958 


1393 


811042 


13 


48 


184()51 


1359 


994857 


33 


189794 


1391 


810206 


12 


49 


185466 


1356 


994838 


33 


190629 


1389 


809371 


11 


50 


186280 


1353 


994818 


33 


191462 


1386 


808538 


10 


51 


9.187092 


1351 


9.994798 


33 


9.192294 


1384 


10.807700 


9 


52 


187903 


1348 


994779 


33 


193124 


1381 


806870 


8 


53 


188712 


1346 


994759 


33 


193953 


1379 


806047 


7 


54 


189519 


1343 


994739 


33 


194780 


1376 


805220 


6 


55 


190325 


1341 


994719 


33 


195606 


1374 


804394 


5 


50 


191130 


1338 


994700 


33 


196430 


1371 


803570 


4 


57 


191933 


1336 


994680 


33 


197253 


1369 


802747 


3 


58 


]9.>734 


1333 


994660 


33 


198074 


1366 


801926 


2 


59 


193534 


1330 


994640 


33 


198894 


1364 


801106 


1 


60 


194332 


1328 


994620 


33 


199713 


1361 


800287 


i 


1 1 


Cosine | 




Sine 




Cotang. 




Tang. 1 


M. 



81 Degrees. 







SINES 


AND TANGENTS. (9 De 


grees.) 




Be 


5 


M. 1 


Sine 


D. 


Cosine 


D. 


Tang. 1 


D. 1 


Cotang. 1 









9.194332 


]328 


9.994620 


33 


9.199713 


1361 


10.800287 


60 




1 


195129 


1326 


994600 


33 


200529 


1359 


799471 


59 




2 


195925 


1323 


994580 


33 


201345 


1356 


798655 


58 




3 


196719 


1321 


994560 


34 


202159 


1354 


797841 


57 




4 


197511 


1318 


994540 


34 


202971 


1352 


797029 


56 




5 


198302 


1316 


994519 


34 


203782 


1349 


796218 


55 




6 


199091 


1313 


994499 


34 


204592 


1347 


795408 


54 




7 


199879 


1311 


994479 


34 


205400 


1345 


794600 


53 




8 


200666 


1308 


994459 


34 


206207 


1342 


793793 


52 




9 


201451 


1306 


994438 


34 


2{)7013 


1340 


792987 


51 




10 


202234 


1304 


994418 


34 


207817 


1338 


792183 


50 




11 


9.203317 


1301 


9.994397 


34 


9.208619 


1335 


10.791381 


49 




12 


20;]797 


1299 


9J4377 


34 


209420 


1333 


799580 


48 




13 


201577 


]296 


994357 


34 


210220 


1331 


789780 


47 




14 


2Q5354 


1294 


994336 


34 


211018 


1328 


•/88082 


46 




15 


20;3131 


1292 


994316 


34 


211815 


1326 


788185 


45 




IG 


20G906 


1289 


994295 


34 


212611 


1324 


787389 


44 




17 


207679 


1287 


994274 


35 


213405 


1321 


786595 


43 




18 


208452 


1285 


994254 


35 


214198 


1319 


785802 


42 




19 


209222 


1282 


994233 


35 


214989 


1317 


785011 


41 




20 


209992 


1280 


994212 


35 


215780 


1315 


784220 


40 




21 


9.210760 


1278 


9.994191 


35 


9.216568 


1312 


10.783432 


39 




22 


211526 


1275 


994171 


35 


217356 


1310 


782644 


38 




23 


212291 


1273 


994150 


35 


218142 


1308 


781858 


37 




24 


213055 


1271 


994129 


35 


218926 


1305 


781074 


36 




25 


213818 


1268 


9J4108 


35 


219710 


1303 


780290 . 


35 




25 


214579 


1266 


994087 


35 


220492 


1301 


779508 


34 




27 


215338 


1264 


994066 


35 


221272 


1299 


778728 


33 




28 


216097 


1201 


994045 


35 


222052 


1297 


777948 


32 




29 


216854 


1259 


994024 


35 


222830 


1294 


777170 


31 




30 


217609 


1257 


994003 


35 


223606 


1292 


776394 


30 




31 


9.218333 


1255 


9.993981 


35 


9.224382 


1290 


10.775618 


29 




32 


219116 


1253 


993930 


35 


225156 


1288 


774844 


28 




33 


219868 


1250 


903939 


35 


225929 


1283 


774071 


27 




34 


220618 


1248 


993918 


35 


22()700 


1284 


773300 


26 




35 


221367 


1246 


9938J6 


36 


227471 


1281 


772529 


25 




36 


222115 


1244 


993875 


36 


228239 


1279 


771761 


24 




37 


222861 


1242 


993854 


36 


229307 


1277 


770903 


23 




38 


223606 


1239 


993832 


36 


229773 


1275 


770227 


22 




39 


224349 


1237 


993811 


36 


230539 


1273 


769461 


21 




40 


225092 


1235 


993789 


36 


231302 


1271 


768698 


20 




41 


9.225833 


1233 


9.993768 


36 


9.232065 


1269 


10.767935 


19 




42 


226573 


1231 


993746 


36 


232826 


1267 


767174 


18 




43 


227311 


1228 


993725 


36 


233586 


1265 


766414 


17 




44 


228048 


1226 


993703 


36 


234345 


1262 


765655 


16 




45 


2.28784 


1224 


993681 


36 


235103 


1260 


764897 


15 




46 


229518 


1222 


993680 


36 


235859 


1258 


764141 


14 




47 


230252 


1220 


993638 


36 


236614 


1256 


763386 


13 




48 


230984 


1218 


993616 


36 


237368 


1254 


762632 


12 




49 


231714 


1216 


9,93594 


37 


238120 


1252 


761880 


11 




50 


232444 


1214 


993572 


37 


238872 


1250 


761128 


10 




51 


9.233172 


1212 


9.993550 


37 


9.239622 


1248 


10.760378 


9 




52 


233899 


1209 


9935-28 


37 


240371 


1246 


759629 


8 




53 


234625 


1207 


993505 


37 


241118 


1244 


758882 


7 




54 


235349 


1205 


993484 


37 


241865 


1242 


758135 


6 




55 


236073 


1203 


993462 


37 


242fiH) 


1240 


757390 


5 




56 


236795 


12)1 


993440 


37 


243354 


1238 


756646 


4 




57 


237515 


1199 


993418 


37 


244097 


1236 


755903 


3 




58 


238235 


1197 


993396 


37 


244839 


1234 


755161 


2 




59 


238953 


1195 


993374 


37 


245579 


1232 


754421 


1 




60 


239670 


1193 


993351 


37 


246319 


1230 


753681 









1 Cosine 




Sine 


1 


1 Cotang. 


1 


1 Tang. 


1 M. 





80 Degrees. 



64 




(10 Degrees.) a 


TABLE OF LOGARITHMIC 








Sine 


D. 


Cosine | 


D. 


Tang. 


D. 


Cotang. 









9.239670 


1J03 


9.993351 


37 


9.246319 


1230 


10.7.53681 


60 




1 


240336 


1191 


993329 


37 


247057 


1228 


752943 


59 




2 


241101 


1189 


903307 


37 


247794 


1226 


752-206 


58 




3 


241814 


1187 


993-285 


37 


248530 


1224 


751470 


57 




4 


242526 


1185 


993262 


37 


249264 


1222 


753736 


56 




5 


24'3QM 


1183 


993240 


37 


249998 


1220 


750002 


55 




C 


243947 


1181 


993217 


38 


250730 


1218 


740270 


54 




7 


2441)56 


1179 


993195 


38 


251461 


1217 


748539 


53 




8 


2453')3 


1177 


993172 


38 


252191 


1215 


747809 


52 




9 


246069 


1175 


993149 


38 


252020 


1213 


747080 


51 




10 


246775 


1173 


993127 


38 


253648 


1211 


746352 


50 




11 


9.247478 


1171 


9.993104 


38 


9.2.54374 


1209 


10.745626 


49 




12 


248181 


1169 


9930H1 


38 


255100 


1207 


744900 


48 




13 


248833 


1167 


993059 


38 


2558-24 


1205 


744176 


47 




14 


249583 


1165 


993036 


38 


256547 


1203 


743453 


46 




15 


25;)28-2 


1163 


993013 


33 


257269 


1201 


74-2731 


45 




16 


250J80 


1161 


99291)0 


38 


257990 


1200 


7J-2010 


44 




17 


25ir,77 


1159 


992967 


38 


258710 


1198 


741290 


43 




18 


252373 


1158 


992944 


38 


259429 


1196 


740571 


42 




19 


2530)7 


1156 


932921 


38 


260146 


1194 


730854 


41 




20 


253761 


1J54 


992898 


38 


260863 


1192 


739137 


40 




21 


9.254451 


1152 


9.992875 


38 


9.261578 


1190 


10.738422 


39 




22 


255144 


1150 


992852 


38 


2{)2292 


1189 


737708 


38 




23 


2.55834 


1148 


992829 


39 


263005 


1187 


736995 


37 




24 


256523 


1146 


99;28a6 


39 


253717 


1185 


736283 


36 




25 


257211 


1144 


992783 


39 


264428 


1183 


735572 


35 




26 


257898 


1142 


992759 


39 


265138 


1181 


734862 


34 




27 


258583 


1141 


992736 


39 


265847 


1179 


734153 


33 




28 


259268 


1139 


992713 


39 


266555 


1178 


733445 


32 




29 


2,59951 


1137 


992699 


39 


267-261 


1176 


732739 


31 




30 


260633 


1135 


992666 


39 


267967 


1174 


732033 


30 




31 


9.261314 


1133 


9.992643 


39 


9.238671 


1172 


10.731329 


29 




32 


2619J4 


1131 


992619 


39 


269375 


1170 


730625 


28 




33 


262673 


1130 


992593 


39 


270077 


1169 


729923 


27 




34 


263351 


1128 


992572 


39 


270779 


1167 


729221 


26 




35 


264027 


1126 


992549 


39 


271479 


1165 


728521 


25 




36 


264703 


1124 


992525 


39 


272178 


1164 


727822 


24 




37 


265377 


1122 


992501 


39 


272876 


1162 


7271-24 


23 




38 


266051 


1120 


992478 


40 


273573 


1160 


726427 


22 




39 


266723 


11J9 


992454 


40 


274-269 


1158 


725731 


21 




41) 


267395 


1117 


992430 


40 


274964 


1157 


725036 


20 




41 


9.268065 


1115 


9.992406 


40 


9.275658 


1155 


10.724342 


19 




42 


268734 


1113 


992382 


40 


276351 


1153 


7-23649 


18 




43 


269402 


11)1 


992359 


40 


277043 


1151 


722957 


17 




44 


270069 


1110 


992335 


40 


277734 


11.50 


7-22266 


16 




45 


270735 


1108 


992311 


40 


278424 


1148 


721576 


15 




46 


271400 


1106 


992287 


40 


279113 


1147 


72J837 


14 




47 


272064 


1L)5 


992263 


40 


279801 


1145 


720199 


13 




48 


272726 


1193 


992239 


40 


280433 


1143 


719512 


12 




49 


273388 


11)1 


992214 


40 


281174 


1141 


7188-26 


11 




50 


274049 


1099 


992190 


49 


281858 


1140 


718142 


10 




51 


9.274708 


1098 


9.992166 


49 


9.28-2542 


1138 


10.717458 


9 




52 


275367 


1096 


992142 


40 


283225 


1136 


716775 


8 




53 


276024 


1094 


9921 17 


41 


2839J7 


1135 


7ir>093 


7 




54 


276681 


1092 


9J2093 


41 


284.583 


1133 


715412 


6 




55 


9773:J7 


1091 


9320^.9 


41 


285-268 


1131 


714732 


5 




56 


277991 


1089 


992044 


41 


285947 


1130 


714053 


4 




57 


278644 


1087 


93202J 


41 


286624 


1128 


713376 


3 




58 


279297 


1086 


9919:)6 


41 


287301 


11-26 


712699 


o 




59 


279948 


1084 


991971 


41 


287977 


11-2,5 


7120*23 


T 




60 


280599 


1082 


991947 


41 


288652 


1123 


711348 







!■ 


Cosine 




Sine 


i 


1 Cotang. 




Tang. 


M. 





79 Degrees 







SINES 


AND TANGENTS. (11 De^eCS.) 




5c 


> 


M. 


1 Sine 


1 D. 


1 Cosine 


1 D. 


1 Tang. 


1 ». 


1 Cotang. 


1 







9.280599 


1082 


9.991947 


41 


9.288652 


1123 


10.711348 


60 




1 


281248 


1081 


991922 


41 


289326 


1122 


710674 


59 




2 


281897 


1079 


991897 


41 


289999 


1120 


710001 


58 




3 


282544 


1077 


991873 


41 


290671 


1118 


709329 


57 




4 


283 19 J 


1076 


991848 


41 


291342 


1117 


708658 


56 




5 


283830 


1074 


991823 


41 


292013 


1115 


707987 


55 




() 


284480 


1072 


991799 


41 


292682 


1114 


707318 


54 




7 


285124 


1071 


991774 


42 


293350 


1112 


700650 


53 




8 


285706 


1069 


991749 


42 


294017 


nil 


705983 


52 




9 


280408 


1067 


991724 


42 


294084 


1109 


705310 


51 




10 


287048 


1006 


991699 


42 


295349 


1107 


704651 


50 




11 


9.287087 


1034 


9.991674 


42 


9.296013 


1106 


10.703987 


49 




12 


288326 


1003 


991049 


42 


296677 


1104 


703323 


48 




13 


28S964 


1001 


991024 


42 


297339 


1103 


702661 


47 




14 


28960J 


i05J 


991599 


42 


298001 


1101 


701999 


46 




15 


2J023a 


1058 


991574 


42 


2980(52 


1100 


70^338 


45 




16 


290870 


1056 


991549 


42 


299322 


1098 


700678 


44 




17 


291504 


1054 


991524 


42 


299980 


lOOo 


700020 


43 




18 


292137 


1053 


991498 


42 


300038 


1095 


699362 


42 




19 


292708 


1051 


991473 


42 


301295 


1093 


698705 


41 




20 


2-93399 


1050 


991448 


42 


301951 


1092 


698049 


40 




21 


9.294029 


1048 


9.991422 


42 


9.302007 


1090 


10.697393 


39 




22 


294058 


1046 


991397 


42 


303261 


1089 


696739 


38 




23 


295286 


1045 


991372 


43 


303914 


1087 


696086 


37 




24 


295913 


1043 


991346 


43 


304567 


1086 


695433 


36 




25 


296539 


1042 


991321 


43 


305218 


1084 


694782 


35 




26 


297104 


1040 


991295 


43 


305869 


1083 


694131 


34 




27 


297788 


1039 


991270 


43 


.306519 


1081 


693481 • 


33 




; 28 


298412 


1037 


991244 


43 


307108 


1080 


692832 


32 




29 


299034 


1036 


991218 


43 


307815 


1078 


692185 


31 




30 


299655 


1034 


9J1193 


43 


308463 


1077 


691537 


30 




31 


9.300276 


1032 


9.991167 


43 


9.309109 


1075 


10.090391 


29 




32 


300895 


1031 


991141 


43 


309754 


1074 


690246 


28 




33 


301514 


1029 


991115 


43 


310398 


1073 


689602 


27 




34 


302132 


1028 


991090 


43 


311042 


1071 


688958 


26 




35 


302748 


102,3 


991004 


43 


311685 


1070 


688315 


25 




36 


303364 


1025 


991038 


43 


312327 


1068 


687673 


24 




37 


303979 


1023 


991012 


43 


312907 


1067 


687033 


23 




38 


304593 


1022 


990983 


43 


313008 


1065 


686392 


22 ■ 




39 


305207 


1020 


990900 


43 


314247 


1064 


685753 


21 




40 


305819 


1019 


990934 


44 


3148S5 


1062 


685115 


20 




41 


9.306430 


1017 


9.990908 


44 


9.315523 


1061 


10.684477 


19 




42 


307041 


1016 


990882 


44 


310159 


1000 


683841 


18 




43 


307650.. 


1014 


990855 


44 


310795 


1058 


683205 


17 




44 


308259 


1013 


990829 


44 


317430 


1057 


682570 


16 




45 


308867 


1011 


990803 


44 


318004 


1055 


681936 


15 




46 


309474 


1010 


990777 


44 


318097 


1054 


681303 


14 




47 


310080 


1008 


990750 


44 


319329 


1053 


680671 


13 




48 


310685 


1007 


990724 


44 


319901 


1051 


680039 


12 




49 


311289 


1005 


990097 


44 


320592 


1050 


679408 


11 




50 


311893 


1004 


'990G71 


44 


321222 


1048 


678778 


10 




51 


9.312495 


1003 


9.990044 


44 


9.321851 


1047 


10.678149 


9 




52 


313097 


1001 


990018 


44 


322479 


1045 


677521 


8 




53 


313698 


1000 


990591 


44 


323100 


1044 


676894 


7 




54 


314297 


998 


990565 


44 


323733 


1043 


676267 


6 




55 


314897 


997 


990538 


44 


324358 


1041 


675642 


5 




56 


315495 


996 


990511 


45 


324983 


1040 


675017 


4 




57 


316092 


994 


990485 


45 


325007 


103<» 


674.393 


3 




58 


316689 


993 


990458 


45 


320231 


1037 


673769 


2 




59 


317284 


991 


990431 


45 


320853 


1030 


673147 


1 




60 


317879 


990 


990404 


45 


327475 


1035 


G72525 







1 


Cosine | 


1 


Sine 1 


1 


Cotang. 


1 


Tang. 1 


"m7 










78 


Degrc 


es. 











56 


( 


12 Degrees.) a 


TABLE OF LOGARITHMIC 






|m. 1 


Sine 1 


D 1 


Cosine | 


D. 


Tang. 1 


D. 1 


Cotang. 1 






'"o 


9.317879 


990 


9.990404 


45 


9.327474 


1035 


10.672526 


60 




1 


318473 


988 


990378 


45 


328095 


1033 


671905 


59 




2 


319066 


987 


990351 


45 


328715 


1032 


671285 


58 




3 


319658 


986 


990324 


45 


329334 


1030 


670666 


57 




4 


320249 


984 


990297 


45 


329953 


1029 


670047 


56 




5 


320840 


983 


990270 


45 


330570 


1028 


669430 


55 




G 


321430 


982 


990243 


45 


331187 


1026 


668813 


54 




7 


322019 


980 


990215 


45 


331803 


1025 


668197 


53 




8 


322607 


979 


990188 


45 


332418 


1024 


667582 


52 




9 


323194 


977 


990161 


45 


333033 


1023 


666967 


51 




10 


323780 


976 


990134 


45 


333646 


1021 


666354 


50 




11 


9.324366 


975 


9.990107 


46 


9.334259 


1020 


10.665741 


49 




12 


324950 


973 


990079 


46 


334871 


1019 


665129 


48 




13 


325534 


972 


990052 


46 


335482 


1017 


604518 


47 




14 


326117 


970 


990025 


46 


336093 


1016 


663907 


46 




15 


326700 


969 


989997 


46 


336702 


1015 


663298 


45 




16 


327281 


968 


989970 


46 


337311 


1013 


662(589 


44 




17 


327862 


966 


989942 


46 


337919 


1012 


662081 


43 




18 


328442 


965 


989915 


46 


338527 


lOlJ 


661473 


42 




19 


329021 


964 


989887 


46 


339133 


1010 


600867 


41 




20 


329599 


962 


989860 


46 


339739 


1008 


6G0261 


40 




21 


9.330176 


961 


9.989832 


4r, 


9.340344 


1007 


10.659656 


39 




22 


330753 


960 


989804 


46 


340948 


1006 


659052 


38 




23 


331329 


958 


989777 


46 


341552 


1004 


658448 


37 




24 


331903 


957 


989749 


47 


342155 


1003 


657845 


3S 




25 


332478 


956 


989721 


47 


342757 


1002 


657243 


35 




26 


333051 


954 


989693 


47 


343358 


1000 


b56642 


34 




27 


333624 


953 


9896('5 


47 


343958 


999 


656042 


33 




28 


334195 


952 


989637 


47 


344558 


998 


655442 


32 




29 


334766 


950 


989609 


47 


345157 


997 


654843 


31 




30 


335337 


949 


989582 


47 


345755 


996 


654245 


30 




31 


9.335906 


948 


9.989553 


47 


9.346353 


994 


10.653647 


29 




32 


336475 


946 


989525 


47 


346949 


993 


C53051 


28 




33 


337043 


945 


989497 


47 


347545 


992 


652455 


27 




34 


337610 


944 


989469 


47 


348141 


991 


651859 


26 




35 


338176 


943 


989441 


47 


348735 


990 


651265 


25 




36 


338742 


941 


989413 


47 


349329 


988 


650671 


24 




37 


339306 


940 


989384 


47 


349922 


987 


650078 


23 




38 


339871 


939 


989356 


47 


350514 


986 


649486 


22 




• 39 


340434 


937 


989328 


47 


351106 


985 


648894 


21 




40 


340996 


936 


989300 


47 


351697 


983 


648303 


20 




41 


9.341558 


935 


9.989271 


47 


9.352287 


982 


10.647713 


19 




42 


342119 


934 


989243 


47 


352876 


981 


647124 


18 




43 


342079 


932 


989214 


47 


353465 


980 


^ 646535 


17 




44 


343239 


931 


989186 


47 


354053 


979 


* 645947 


16 




45 


343797 


930 


989157 


47 


354640 


977 


645360 


15 




46 


344355 


929 


989128 


4d 


355227 


976 


644773 


14 




47 


344912 


927 


989100 


48 


35.'>813 


975 


644187 


13 




48 


345409 


926 


989071 


48 


356398 


974 


643602 


12 




49 


346024 


925 


989042 


48 


356982 


973 


643018 


11 




50 


346579 


924 


989014 


48 


357566 


971 


642434 


10 




51 


9.347134 


922 


9.988985 


48 


9.358149 


970 


10.641851 


9 




52 


347687 


921 


988956 


48 


358731 


969 


641269 


8 




53 


348240 


920 


988927 


48 


359313 


968 


640687 


7 




54 


348792 


919 


988898 


48 


359893 


907 


64U107 


6 




55 


349343 


917 


988869 


48 


360474 


966 


639526 


5 




56 


349893 


916 


988840 


48 


301053 


965 


63-947 


4 




57 


350443 


915 


988811 


49 


361632 


963 


638308 


3 




58 


350992 


914 


988782 


49 


362210 


902 


637790 


2 




59 


351540 


913 


988753 


49 


3r)2787 


901 


637213 


1 




60 


352U86 


911 


988724 


49 


363364 


960 


636636 







1 Cosine 


1 


1 Sine 


1 


1 Cotang. 


1 


1 Tang. 


1 M. 




MBHlMi 






77 


Deg] 


:eo6. 















SINES 


AND TANGENTS. (13 Degrees.) 


5' 


7 


M. 


Sine 


D. 


1 Cosine 


1 D. 


1 Tang. 


D. 


1 Cotang. 


I 







9.352088 


911 


9.938724 


49 


9.363364 


960 


10.636636 


60 




] 


352635 


910 


988695 


49 


3G3940 


959 


636060 


59 




2 


353181 


909 


988666 


49 


364515 


958 


635485 


58 




3 


353726 


908 


988630 


49 


365090 


957 


634910 


57 




4 


354271 ' 


907 


988607 


49 


365664 


955 


634336 


56 




5 


354815 


905 


988578 


49 


366237 


954 


633763 


55 




6 


355358 


904 


988548 


49 


366810 


953 


6.33190 


54 




7 


355901 


903 


988519 


49 


367382 


952 


632618 


53 




8 


356443 


9!)2 


988489 


49 


367953 


951 


632047 


52 




9 


356984 


901 


988460 


49 


368524 


950 


63 1476 


51 




10 


357524 


899 


988430 


49 


369094 


949 


630906 


50 




n 


9.358064 


898 


9.988401 


49 


9.369633 


948 


10.63)337 


49 




12 


358603 


897 


988371 


49 


37U232 


946 


G2J768 


48 




13 


359141 


896 


933342 


49 


370799 


945 


62920] 


47 




14 


359678 


895 


938312 


50 


371367 


914 


628633 


46 




J5 


300215 


893 


938282 


50 


371933 


943 


623067 


45 




J6 


360752 


892 


988252 


50 


372499 


942 


627.501 


44 




1 17 


361287 


891 


983223 


50 


373064 


941 


626936 


43 




18 


331822 


890 


988193 


50 


373629 


940 


626371 


42 




! li^ 


362356 


889 


938163 


59 


374193 


939 


625807 


41 




20 


362889 


888 


988133 


50 


374756 


938 


625244 


40 




21 


9.363422 


887 


9.938103 


50 


9.375319 


937 


10.624681 


39 




22 


363954 


885 


988073 


50 


375881 


935 


624119 


38 




23 


364485 


884 


988043 


50 


376442 


934 


623558 


37 




24 


365016 


883 


938013 


50 


377003 


933 


622997 


36 




25 


3C5546 


882 


937983 


50 


377563 


932 


622437 


35 




26 


366075 


881 


987953 


50 


378122 


931 


621878 


34 




27 


366604 


880 


987922 


50 


378681 


930 


621319 


33 




28 


367131 


879 


987892 


50 


379239 


929 


620761 


32 




20 


367659 


877 


987862 


50 


379797 


9-28 


620203 


31 




30 


368185 


876 


987832 


51 


380354 


927 


619646 


30 




31 


9.36871] 


875 


9.987801 


51 


9.380910 


920 


10.619090 


29 




32 


369236 


874 


987771 


51 


381466 


925 


618534 


28 




33 


369761 


873 


987740 


51 


382020 


924 


617980 


27 




34 


370285 


872 


987710 


51 


382575 


923 


617425 


26 




35 


370808 


871 


987679 


51 


383 J 29 


922 


616871 


25 




36 


371330 


870 


987649 


51 


383682 


921 


616318 


24 




37 


371852 


839 


987618 


51 


384234 


920 


615766 


23 




38 


372373 


867 


987588 


51 


384786 


919 


615214 


22 




39 


372894 


866 


987557 


51 


385337 


918 


614663 


21 




40 


373414 


865 


987528 


51 


385888 


917 


614112 


90 




41 


9.373933 


864 


9.987493 


51 


9.386438 


915 


19.613562 


19 




42 


374452 


863 


987465 


51 


386987 


914 


613013 


18 




43 


374970 


862 


987434 


51 


387536 


913 


612464 


17 




44 


375487 


861 


937403 


52 


388084 


912 


611916 


16 




45 


376003 


8G0 


987372 


52 


388631 


911 


611369 


15 




46 


376519 


859 


937341 


52 


389178 


910 


610822 


14 




47 


377035 


858 


937310 


52 


389724 


909 


610276 


13 




48 


377549 


857 


937279 


52 


390270 


908 


609730 


12 




49 


378063 


856 


987248 


52 


390815 


907 


609185 


11 




50 


378577 


854 


. 987217 


52 


391360 


906 


608640 


10 




51 


9.379089 


853 


9.987186 


52 


9.391903 


905 


10.608097 


9 




52 


379601 


852 


987155 


52 


392447 


904 


607553 


8 




53 


380113 


851 


987124 


52 


392989 


903 


607011 


7 , 




54 


380624 


850 


987092 


52 


393531 


902 


606469 


6 1 




55 


381134 


849 


987061 


52 


394073 


901 


605927 


5 1 




56 


381643 


843 


987030 


52 


394614 


900 


605386 


4 




57 


382152 


847 


986998 


52 


395154 


899 


604846 


3 1 




58 


382661 


846 


986967 


52 


395694 


898 


604306 


2 ! 




59 


383168 


845 


986936 


52 


396233 


897 


603767 


1 1 




60 


383675 


844 


986904 


52 


396771 


896 


603229 


i 




1 


Cosine | 




Sine 




Cotang. 1 


1 


Tang. 


M. 





76 Degrees 



58 



(14 Degrees.) a table op logarithmic 



l>I. 


1 Sine 


1 D 


1 Cosine | D. 


1 Tang. 


1 D. 


1 Cotang. 


1 
60 




1 


9.383675 


844 


9 986904 


52 


9.396771 


896 


10.603229 




1 


384182 


843 


986873 


53 


397309 


896 


G02691 


59 




o 


384G87 


842 


980841 


53 


397846 


895 


602154 


58 




:1 


385192 


841 


980809 


53 


398383 


894 


601617 


57 




, 4 


385C97 


840 


986778 


53 


398919 


893 


601081 


56 




5 


380201 


839 


986746 


53 


399455 


892 


600545 


55 




6 


386704 


838 


986714 


53 


399990 


891 


600010 


54 




7 


387207 


837 


986683 


53 


400524 


890 


599476 


53 




8 


387709 


830 


986651 


53 


401058 


889 


598942 


52 




9 


388210 


835 


986619 


53 


401591 


883 


598409 


51 




10 


388711 


834 


98G587 


53 


402124 


887 


597876 


50 




IJ 


9.389211 


833 


9 986555 


53 


9.402656 


886 


10.597344 


49 




12 


389711 


832 


986523 


53 


403187 


885 


596813 


48 




13 


390210 


831 


986491 


53 


403718 


884 


596282 


47 




J4 
15 


390708 


830 


986459 


53 


404249 


883 


595751 


46 




391206 


828 


986427 


53 


404778 


882 


595222 


45 




IG 


391703 


827 


986395 


53 


405308 


881 


594692 


44 




17 


392199 


826 


986363 


54 


40.5836 


880 


594164 


43 




18 


39-2C95 


825 


986331 


54 


406304 


879 


593636 


42 




19 


393191 


824 


986299 


54 


405892 


878 


593108 


41 




20 


393085 


823 


983266 


54 


407419 


877 


592581 


40 




21 


9.394179 


822 


9 986234 


54 


9.407945 


876 


10.592055 


39 




22 


394G73 


821 


980202 


54 


408471 


875 


591529 


38 




23 


395166 


820 


986169 


54 


408097 


874 


591003 


37 




24 


395658 


819 


986137 


54 


409521 


874 


59;)479 


36 




25 


396150 


818 


986104 


54 


410045 


873 


589955 


35 




2G 


39G641 


817 


980072 


51 


410509 


872 


589431 


34 




27 


397132 


817 


980039 


54 


411092 


871 


588908 


33 




28 


397621 


816 


986007 


54 


411615 


870 


588385 


32 




29 


398111 


815 


985974 


54 


412137 


869 


587863 


31 




30 


398600 


814 


985942 


54 


412658 


868 


587342 


30 




31 


9.399088 


813 


9.985909 


55 


9.413179 


867 


10.586821 


29 




32 


399575 


8l2 


985876 


55 


413699 


866 


580301 


28 




33 


400U62 


811 


985843 


55 


414219 


8f)5 


585781 


27 




34 


400549 


813 


985811 


55 


414738 


864 


585262 


26 




35 


401035 


8J9 


985778 


55 


415257 


8()4 


584743 


25 




36 


401520 


8dS 


985745 


55 


415775 


863 


584225 


24 




37 


402005 


807 


985712 


55 


416293 


80)2 


583707 


23 , 




38 


402189 


803 


985679 


55 


416810 


861 


583190 


O.) 




39 


402972 


805 


985646 


55 


417323 


860 


5820)74 


21 




40 


403455 


804 


985613 


55 


417842 


859 


582158 


20 




41 


9.403938 


803 


9.985,580 


55 


9.41835.-] 


858 


10.. 5816 42 


19 




42 


404420 


802 


985547 


55 


418873 


857 


581 127 


18 




43 


404901 


801 


985514 


55 


419387 


856 


580013 


17 




44 


405382 


80i) 


985480 


55 


419901 


855 


580099 


16 




45 


405862 


799 


985447 


55 


420415 


855 


579585 


15 




46 


406341 


798 


985414 


56 


420927 


854 


579073 


14 




47 


406820 


797 


985380 


56 


421440 


853 


578560 


13 




48 


407299 


796 


985347 


56 


421952 


852 


578048 


12 




49 


407777 


795 


985314 


56 


422403 


851 


577537 


11 




50 


408254 


794 


985280 


56 


422974 


850 


577026 


10 




51 


9.408731 


794 


9.985247 


56 


9.423484 


849 


10., 576516 


9 




52 


409207 


793 


985213 


56 


423993 


848 


576007 


8 




53 


409(582 


792 


985180 


56 


424503 


848 


575497 


7 




54 


410157 


791 


985146 


56 


425011 


847 


574989 


6 




55 


4106)32 


790 


985113 


56 


42.5519 


846 


574481 


5 




56 


411106 


789 


985079 


56 


426027 


845 


573973 


4 




57 


411579 


788 


985045 


5) 


426534 


844 


573466 


3 




58 


412052 


787 


98501 1 


56 


427041 


843 


572959 


2 




59 


412524 


786 


984978 


56 


427.547 


843 


.57^2-153 


1 




60 


412996 


785 


984944 


56 


428052 


842 


571948 







1T[ 


Cosine | 


I 


Sine 1 1 


Cotang. 1 


1 


Tang. 1 


"»iT 










75 


Degi 


ees 











SINES AND TANGENTS. (15 Degrees.) 



59 



M 


, Sine 


1 !>• 


Cosine 


1 !>• 


Tang. 


1 »• 


1 Cotang. 


1 





9.412996 


785 


9.984944 


57 


9.428052 


842 


10.571948 


60 


1 


413467 


784 


984910 


57 


428557 


841 


571443 


59 


2 


413938 


783 


984876 


57 


420062 


840 


570938 


58 


3 


414408 


783 


984842 


57 


429566 


839 


570434 


57 


4 


414878 


782 


984808 


57 


430070 


838 


569930 


56 


5 


415347 


781 


984774 


57 


430573 


838 


569427 


55 


6 


415815 


780 


984740 


57 


431075 


^37 


568925 


54 


7 


416283 


779 


984706 


57 


431577 


836 


568423 


53 


8 


410751 


778 


984672 


57 


432079 


835 


567921 


52 


9 


417217 


777 


984637 


57 


432580 


834 


567420 


51 


10 


417684 


776 


984603 


57 


433080 


833 


566920 


50 


11 


9.418150 


775 


9.984569 


57 


9.433580 


832 


10.566420 


49 


12 


418615 


774 


984535 


57 


434080 


832 


565920 


48 


13 


419079 


773 


984500 


57 


434579 


831 


565421 


47 


14 


419544 


773 


984466 


57 


435078 


830 


. 564922 


46 


15 


420007 


772 


984432 


58 


435576 


829 


564424 


45 


16 


420470 


771 


984397 


58 


4 36073 


828 


563927 


44 


17 


420933 


770 


984363 


58 


436570 


828 


563430 


43 


18 


421.395 


769 


984328 


58 


437067 


827 


562933 


42 


19 


421857 


768 


984294 


58 


437563 


826 


562437 


41 


20 


422318 


767 


981259 


58 


438059 


825 


561941 


40 


21 


9.422778 


767 


9.984224 


58 


9.438554 


824 


10.561446 


39 


22 


423238 


766 


984190 


58 


43-9048 


823 


560952 


38 


23 


423697 


705 


984155 


58 


439543 


823 


560457 


37 


24 


424156 


764 


984120 


58 


440036 


822 


559964 


36 


25 


424615 


763 


984085 


58 


440529 


82i 


559471 


35 


26 


425073 


762 


984050 


58 


441022 


820 


558978 


34 


27 


425530 


761 


984015 


58 


441514 


819 


558486 


33 


28 


425987 


760 


983981 


58 


442006 


819 


557994 


32 


29 


426443 


700 


983946 


58 


442497 


818 


557503 


31 


30 


426899 


759 


983911 


58 


442988 


817 


557012 


30 


31 


9.427354 


758 


9.983875 


58 


9.443479 


816 


10.556521 


29 


?2 


427809 


757 


983840 


59 


443908 


816 


556032 


28 


33 


428263 


756 


983805 


59 


444458 


8J5 


555542 


27 


34 


428717 


755 


983770 


59 


444947 


814 


555053 


26 


35 


429170 


754 


983735 


59 


445435 


813 


554565 


25 


36 


429323 


753 


983700 


59 


445923 


812 


554077 


24 


37 


430075 


752 


983664 


59 


446411 


812 


553589 


23 


38 


430527 


7.>2 


983629 


59 


440898 


811 


553102 


22 


39 


430978 


751 


983594 


59 


447384 


810 


5.52616 


21 \ 


40 


431429 


750 


983558 


59 


447870 


809 


552130 


20 


41 


9.431879 


749 


9.983523 


59 


9.448356 


809 


10.551644 


19 


42 


432329 


749 


983487 


59 


448841 


808 


551159 


18 


43 


432778 


748 


983452 


59 


449326 


807 


550674 


17 


44 


433226 


747 


983416 


59 


449810 


806 


550190 


16 


45 


433675 


746 


983381 


59 


450294 


806 


549706 


15 


46 


434122 


745 


983345 


59 


450777 


805 


549223 


14 


47 


434569 


744 


933309 


59 


451260 


804 


548740 


13 


48 


435016 


744 


983273 


60 


451743 


803 


548257 


12 


49 


435482 


743 


983238 


00 


45222.5 


802 


547775 


11 


50 


435908 


742 


983202 


CO 


452706 


802 


547294 


10 


51 


9.436.3.53 


741 


9.983166 


CO 


9.453187 


801 


10.546813 


9 


52 


436798 


740 


983130 


60 


453668 


800 


546332 


8 


53 


437242 


740 


983094 


60 


454148 


799 


545852 


7 


54 


437086 


739 


983058 


60 


454628 


799 


545372 


6 


55 


438129 


738 


983022 


60 


455107 


798 


544893 


5 


56 


438572 


737 


982986 


60 


455586 


797 


544414 


4 


57 


439014 


736 


982950 


60 


456064 


796 


543936 


3 


58 


439456 


736 


982914 


60 


456542 


796 


543458 


2 


59 


439897 


735 


982878 


60 


457019 


795 


542981 


1 


60 


440338 


734 


982842 


60 


457496 


794 


542504 





1 


Cosine | 




Sine 


1 


Cotang. 1 




Tang. 1 


M. 



74 Degrees 





60 




[16 Degrees.) a 


TABLE OF LOGARITHMIC 






M. 


1 Sine 


1 »> 


1 Cosine 


1 D 


I Tang. 


I D 


Cotang. 


1 







9.440338 


734 


9.982842 


60 


9.457496 


794 


10.542504 


60 




1 


440778 


733 


982805 


60 


457973 


793 


542027 


59 




2 


4412J8 


732 


982769 


61 


458449 


793 


541551 


58 




3 


441658 


731 


982733 


61 


458925 


792 


541075 


57 




4 


442096 


731 


982696 


61 


459400 


791 


540600 


56 




5 


442535 


730 


982660 


61 


459875 


790 


540125 


55 




6 


442973 


729' 


982624 


61 


460349 


790 


539651 


54 




7 


443410 


728 


982587 


61 


460823 


789 


539177 


53 




8 


443847 


727 


982551 


61 


461297 


788 


538703 


52 




9 


444284 


727 


982514 


61 


461770 


788 


538230 


51 




10 


444720 


726 


982477 


61 


462242 


787 


537758 


50 1 




11 


9.445155 


725 


9.982441 


61 


9.462714 


786 


10.537286 


49 




12 


445590 


724 


982404 


61 


463186 


785 


536814 


48 




13 


446025 


723 


982367 


61 


463658 


785 


536.342 


47 




14 


44645tt 


723 


982331 


61 


464129 


784 


535871 


46 




15 


446893 


722 


982294 


61 


464599 


783 


53.5401 


45 




m 


447326 


721 


982257 


61 


465069 


783 


534931 


44 




17 


447759 


720 


982220 


62 


46.5539 


782 


534461 


43 




18 


448191 


720 


982183 


62 


466008 


781 


533992 


42 




19 


448623 


719 


982146 


62 


466476 


780 


533524 


41 




20 


449054 


718 


982109 


62 


466945 


780 


533055 


40 




21 


9.449485 


717 


9.982072 


62 


9.467413 


779 


10.532587 


39 




22 


449915 


716 


982035 


62 


467880 


778 


532120 


38 




23 


450345 


716 


981998 


62 


468347 


778 


531653 


37 




24 


450775 


715 


981961 


62 


468814 


777 


531186 


36 




25 


451204 


714 


981924 


62 


469280 


776 


530720 


35 




26 


451632 


713 


981886 


62 


469746 


775 


530254 


34 




27 


452060 


713 


981849 


62 


470211 


775 


529789 


33 




28 


45.2488 


712 


981812 


62 


470676 


774 


529324 


32 




29 


452915 


711 


981774 


62 


471141 


773 


528859 


31 




30 


453342 


710 


981737 


62 


471605 


773 


528395 


30 




31 


9.453768 


710 


9.981699 


63 


9.472068 


772 


10.527932 


29 




32 


454194 


709 


981662 


63 


472532 


771 


527468 


28 




33 


454619 


708 


981625 


63 


472995 


771 


52.005 


27 




34 


455044 


707 


981587 


63 


473457 


770 


52G543 


26 




35 


455469 


707 


981549 


63 


473919 


769 


526081 


25 




36 


455893 


706 


981512 


63 


474381 


769 


525619 


24 




37 


450316 


705 


981474 


63 


474842 


768 


525158 


23 




38 


456739 


704 


981436 


63 


475303 


767 


524697 


22 




39 


457162 


704 


981399 


63 


475763 


767 


524237 


21 




40 


457584 


703 


981361 


63 


476223 


766 


523777 


20 




41 


9.458006 


702 


9.981.323 


63 


9.476683 


765 


10.523317 


19 




42 


458427 


701 


981285 


63 


477142 


765 


522858 


18 




43 


458848 


701 


981247 


63 


477601 


764 


522399 


17 




44 


459268 


700 


981209 


63 


478059 


763 


521941 


16 




45 


459088 


699 


981171 


63 


478517 


763 


521483 


15 




46 


460108 


698 


981133 


64 


478975 


762 


521025 


14 




47 


460527 


698 


981095 


64 


479432 


761 


520568 


13 




48 


460946 


697 


981057 


64 


479889 


761 


520111 


12 




49 


461364 


696 


981019 


64 


480345 


760 


519655 


11 




50 


461782 


695 


980981 


64 


480801 


759 


519199 


10 




51 


9.462199 


095 


9.980942 


64 


9.481257 


759 


10.518743 


9 




52 


462616 


694 


980904 


64 


481712 


758 


518288 


8 




53 


463032 


693 


980866 


64 


482167 


757 


517833 


7 




54 


463448 


693 


980827 


64 


482621 


757 


517379 


6 




55 


463864 


692 


980789 


64 


483075 


756 


516925 


5 




56 


464279 


691 


980750 


64 


48:i529 


755 


510471 


4 




57 


464694 


690 


980712 


64 


483982 


755 


51(5018 


3 




58 


465108 


690 


980(573 


(54 


484435 


754 


5155(55 


2 




59 


4(55.522 


()89 


980635 


64 


484887 


753 


:>15113 


1 




60 


465935 


688 


980596 


64 


485339 


753 


514661 







1 


Cosine | 


1 


Sine 1 


1 


Cotang. 1 


1 


Tang. 1 


M. 



73 Degrees 







SINES 


AND TANGENTS. (17 Degrees.) 




61 


1 M. 


1 Sine 


1 »• 


( Cosine 


I D. 


1 Tang. 


1 D. 


Cotang. 


1 1 





9.4G5935 


688 


9.980595 


i 64 


9.485339 


■;;.)5 


10.514661 


60 


1 


466348 


688 


930558 


64 


485791 


752 


514-209 


59 


o 


466761 


687 


930519 


65 


486242 


751 


51 3753 


58 


3 


467173 


686 


980480 


65 


486693 


751 


513307 


57 


4 


467585 


685 


980442 


65 


487143 


750 


512857 


56 


5 


487996 


685 


930403 


05 


487593 


749 


512407 


55 


6 


468407 


684 


980364 


65 


488343 


749 


511957 


54 


7 


468817 


683 


983325 


65 


488492 


743 


511508 


53 


8 


469227 


683 


983238 


65 


433941 


747 


511059 


52 


9 


469G37 


682 


980247 


65 


489390 


747 


510310 


51 


10 


470046 


681 


980208 


65 


489838 


746 


510162 


50 


11 


9.470455 


680 


9.980109 


65 


9.490286 


746 


10.509714 


49 


12 


470863 


680 


980130 


65 


490733 


745 


503257 


48 


13 


471271 


679 


980091 


65 


491180 


744 


533820 


47 


14 


471679 


678 


980052 


65 


491627 


744 


508373 


46 


15 


472086 


678 


930012 


65 


492073 


743 


507927 


45 


16 


472492 


677 


979973 


65 


432519 


743 


507431 


44 


17 


472898 


676 


979934 


66 


432335 


742 


507035 


43 


18 


473301 


676 


979395 


68 


433410 


741 


506590 


42 


19 


473710 


675 


979855 


66 


49.3854 


740 


506146 


41 


20 


474115 


674 


979816 


63 


494299 


740 


505701 


40 


21 


9.474519 


674 


9.979776 


66 


9.494743 


740 


10.505257 


39 


22 


474923 


673 


979737 


66 


495188 


739 


504814 


38 


23 


475327 


672 


979697 


66 


495333 


738 


504370 


37 


24 


475730 


672 


979358 


63 


498073 


737 


503927 


36 


25 


476133 


671 


979618 


63 


493515 


737 


503435 


35 


26 


47G533 


670 


979579 


66 


498957 


736 


503043 


34 


27 


476938 


669 


979539 


66 


497399 


736 


502801 


33 


28 


477340 


669 


979493 


63 


497841 


735 


502159 


32 


29 


477741 


668 


979459 


63 


498282 


734 


501718 


31 


30 


478142 


667 


979420 


68 


498722 


734 


501278 


30 


31 


9.478542 


667 


9.979333 


66 


9.499183 


733 


10.. 500837 


29 


32 


478942 


663 


979340 


66 


499303 


733 


500397 


28 


33 


479342 


665 


979330 


67 


500042 


732 


499958 


27 


34 


479741 


635 


979230 


67 


500481 


731 


499519 


26 


35 


480140 


634 


979220 


67 


503920 


731 


493080 


25 


36 


430539 


683 


979130 


67 


501359 


730 


498341 


24 


37 


480^37 


663 


979140 


67 


501797 


730 


498203 


23 


38 


481334 


682 


979100 


67 


502235 


729 


497765 


22 


3J 


48173.1 


631 


979359 


67 


502672 


728 


497328 


21 


40 


482128 


661 


979319 


67 


503109 


728 


498891 


20 


41 


9.482525 


660 


9.973379 


67 


9.503546 


727 


10.496454 


19 


42 


482321 


659 


978339 


67 


503982 


727 


496018 


18 


43 


483316 


659 


978838 


67 


504413 


726 


495582 


17 


44 


483712 


653 


978358 


67 


504854 


725 


495146 


16 


45 


484107 


657 


978817 


67 


505289 


725 


494711 


15 


48 


484501 


657 


978777 


67 


505724 


724 


494276 


14 


47 


484895 


653 


978736 


67 


508159 


724 


433341 


13 


48 


485-289 


655 


978393 


68 


503593 


7-23 


493407 


12 


49 


435382 


655 


978355 


68 


507027 


7-22 


492973 


11 


50 


483075 


654 


978815 


68 


507460 


722 


492540 


10 


51 


9.483467 


653 


9.978574 


68 


9.507893 


721 


10.492107 


9 


52 


483830 


653 


978533 


68 


503323 


721 


491874 


8 


53 


487251 


652 


973493 


68 


508759 


720 


491241 


7 


54 


487643 


651 


973452 


68 


503191 


719 


493809 


6 


55 


483034 


651 


978411 


68 


509322 


713 


493373 


5 


56 


488424 


650 


978370 


68 


510054 


718 


439346 


4 


57 


488814 


653 


973329 


68 


510485 


718 


489515 


3 


58 


489204 


649 


978233 


68 


510916 


717 


489084 


2 


59 


489593 


648 


978247 


68 


51-1348 


716 


483854 


1 


60 


489982 


648 


978206 


68 


51 1776 


716 


488224 





1 


(Josine | 


1 


Sine 1 


1 


Cotang. 1 


1 


Tang. 1 


M. 1 



72 Degrees. 



62 



(18 Degrees.) a table op logarithmic 



M. 




1 Sine 


1 D 


[ Cosine 


1 D. 


1 Tang. 


! D. 


1 Cotang. 


i 




9 489982 


648 


9.978206 


68 


9.511776 


716 


10.488224 


60 




1 


490371 


648 


978165 


68 


512206 


716 


487794 


59 




2 


490759 


647 


978124 


68 


512G35 


715 


487365 


58 




3 


491147 


646 


978083 


69 


513064 


714 


486936 


57 




4 


491535 


646 


978042 


69 


513493 


714 


486507 


56 




5 


491922 


645 


978001 


69 


513921 


713 


486079 


55 




6 


492308 


644 


977959 


69 


514349 


713 


485651 


54 




7 


492695 


644 


977918 


6J 


514777 


7J2 


485223 


53 




8 


493081 


643 


97:877 


69 


515204 


712 


48479G 


52 




1 9 


4934G6 


642 


977835 


69 


515631 


711 


484369 


51 




10 


493851 


642 


977794 


69 


516057 


710 


483943 


50 




11 


9.494236 


641 


9.977752 


G9 


9.516484 


710 


10.483516 


49 




12 


494621 


641 


977711 


69 


51G910 


709 


483090 


48 




13 


495005 


640 


9776G9 


69 


517335 


709 


482GG5 


47 




14 


495388 


639 


977628 


69 


5] 7761 


708 


482239 


46 




]5 


495772 


639 


977588 


69 


518185 


708 


481815 


45 




IG 


49G154 


638 


977.544 


70 


518610 


707 


481390 


44 




17 


49G537 


637 


977503 


70 


519034 


706 


480966 


43 




18 


496919 


637 


977461 


70 


519458 


706 


480542 


42 




19 


497301 


636 


977419 


70 


519882 


705 


480118 


41 




20 


497682 


636 


977377 


70 


520305 


705 


479G95 


40 




21 


9.4980G4 


635 


9.977335 


70 


9.520728 


704 


10.479272 


39 




22 


498444 


C34 


977293 


70 


521151 


703 


478849 


38 




23 


498825 


634 


977251 


70 


521573 


703 


478427 


37 




24 


499204 


633 


977209 


70 


521995 


703 


478005 


36 




25 


499584 


632 


977157 


70 


522417 


702 


477583 


35 




26 


4999G3 


632 


977125 


70 


522838 


702 


477162 


34 




27 


500342 


631 


977083 


70 


523259 


701 


476741 


33 




28 


500721 


631 


977041 


70 


■ 523680 


701 


47G320 


32 




29 


501099 


630 


976999 


70 


524100 


700 


475900 


31 




30 


501476 


629 


97C957 


70 


524520 


099 


475480 


30 




31 


9.501854 


629 


9.976914 


70 


9.524939 


G99 


10.475061 


29 




32 


502231 


028 


97(5872 


71 


525359 


698 


474G41 


28 




33 


502G07 


628 


976830 


71 


525778 


698 


474222 


27 




34 


502984 


627 


97G787 


71 


52C197 


697 


473803 


26 




35 


5033G0 


C26 


976745 


71 


526C15 


697 


473385 


25 




36 


503735 


626 


976702 


71 


527033 


696 


4729G7 


24 




37 


504110 


625 


976660 


71 


527451 


696 


472.549 


23 




38 


504485 


625 


976617 


71 


527868 


695 


4721.32 


22 




39 


5048G0 


624 


97G574 


71 


528285 


695 


471715 


21 




40 


505234 


623 


976532 


71 


528702 


694 


471298 


20 




41 


9.505008 


623 


9.976489 


71 


9.. 529 119 


693 


10.470881 


19 




42 


505981 


622 


976446 


71 


529535 


G93 


470465 


18 




43 


50G354 


622 


976404 


71 


5299.50 


693 


470050 


17 




44 


50G727 


C21 


97G3G1 


71 


530366 


692 


469634 


16 




45 


5U7099 


620 


976318 


71 


530781 


691 


469219 


15 




46 


507471 


620 


97(5275 


71 


531196 


691 


468804 


14 




47 


507843 


619 


97G232 


72 


531611 


690 


468389 


13 




48 


508214 


619 


976189 


72 


532025 


690 


4(57975 


12 




49 


508585 


618 


976146 


72 


532439 


689 


467561 


11 




50 


508956 


618 


976103 


72 


532853 


689 


467147 


10 




51 


9.509326 


617 


9.9760(50 


72 


9.. 533266 


688 


10.466734 


9 \ 




52 


509:;9!) 


61G 


976017 


72 


533(;79 


688 


4(5(5321 


8 , 




53 


51(>i,)()5 


616 


975974 


72 


534092 


687 


4G5908 


7 




54 


5104.34 


615 


975930 


72 


534504 


687 


4(5,549(5 


6 




55 


510803 


615 


975887 


72 


534916 


686 


465084 


5 




50 


511172 


614 


975844 


72 


535328 


686 


4(54672 


4 




57 


511.540 


613 


9758, to 


72 


535739 


685 


464261 


3 




58 


511907 


613 


975757 


72 


5.36150 


685 


463850 


2 




59 


512275 


612 


975714 


72 


536561 


684 


4(53439 


1 




60 


512642 


612 


975670 


72 


536972 


684 


463028 







1 


Cosine | 


1 


Sine 1 


1 


Cotang. 1 


1 


Tang^ 1 


M. 





71 Degrees 







SINES 


AND TANGENTS. (19 Degrees.^] 




63 




M. 


Sine 


1 !>• 


i Cosine 


1 ^■ 


( Tang 


D 


Cotang. 









9.512642 


612 


9.975670 


73 


9.536972 


684 


10.463028 


60 




1 


513009 


611 


975627 


73 


,537382 


683 


462618 


59 




2 


513375 


611 


975583 


73 


537792 


683 


462208 


58 




3 


5137-il 


010 


975539 


73 


538202 


682 


461798 


57 




4 


514107 


609 


975496 


73 


538611 


682 


4G1389 


56 




5 


514472 


609 


975452 


73 


539020 


681 


460980 


55 




6 


514837 


608 


975408 


73 


539429 


681 


460571 


54 




7 


515202 


608 


975365 


73 


539837 


680 


460163 


53 




8 


515506 


607 


975321 


73 


540245 


680 


459755 


52 




9 


515930 


007 


975277 


73 


540653 


679 


459347 


51 




JO 


516294 


606 


975233 


73 


541061 


679 


458939 


50 




11 


9.516657 


G05 


9.975189 


73 


9.541468 


678 


10.4.58532 


49 




12 


517020 


605 


975145 


73 


541875 


678 


458125 


48 




13 


517382 


604 


975101 


73 


542281 


677 


457719 


47 




14 


517745 


604 


975057 


73 


542688 


677 


457312 


46 




15 


518107 


603 


975013 


73 


543094 


676 


456906 


45 




IG 


518468 


603 


974969 


74 


.543499 


676 


456501 


44 




17 


518829 


602 


974925 


74 


543905 


675 


456095 


43 




18 


519190 


601 


974880 


74 


544310 


675 


4.55690 


42 




19 


519551 


601 


974836 


74 


544715 


674 


455285 


41 




20 


519911 


000 


974792 


74 


545119 


674 


454881 


40 




21 


9.520271 


600 


9.974748 


74 


9.54.5524 


673 


10.454476 


39 




22 


520631 


599 


974703 


74 


545928 


673 


454072 


38 




23 


520990 


599 


974659 


74 


546331 


672 


453669 


37 




24 


521349 


598 


9746H 


74 


546735 


672 


453265 


36 




25 


521707 


598 


974570 


74 


5.i7]38 


671 


452862 


35 




26 


522066 


597 


974525 


74 


547540 


671 


452460 


34 




27 


522424 


596 


9744S1 


74 


547943 


670 


452057 


33 




28 


522781 


596 


974436 


74 


548345 


670 


451655 


32 




29 


523138 


595 


974391 


74 


548747 


669 


4512.53 


31 




30 


523495 


595 


974347 


75 


549149 


669 


450851 


30 




31 


9.523852 


594 


9.974302 


75 


9.549550 


068 


10.450450 


29 




32 


524208 


594 


974257 


75 


549951 


668 


450049 


28 




33 


524564 


593 


974212 


75 


550352 


667 


449648 


27 




34 


524920 


593 


974167 


75 


550752 


C67 


449248 


26 




35 


525275 


592 


974122 


75 


551152 


C66 


448848 


25 




30 


525630 


591 


974077 


75 


551552 


666 


448448 


24 




37 


525984 


591 


974032 


75 


551952 


665 


448048 


23 




38 


526339 


590 


973987 


75 


552351 


665 


447649 


22 




39 


52(]693 


590 


973942 


75 


552750 


665 


447250 


21 




40 


527046 


589 


973897 


75 


553149 


664 


446851 


20 




41 


9.527400 


589 


9.973852 


75 


9.553548 


664 


10.446452 


19 




42 


527753 


588 


973807 


75 


553946 


663 


446054 


18 




43 


528105 


588 


973761 


75 


554344 


663 


445656 


17. 




44 


528458 


587 


973716 


76 


554741 


662 


445259 


16 




45 


528810 


587 


973671 


76 


555139 


662 


444861 


15 




4G 


529161 


586 


973625 


76 


555536 


C61 


444464 


14 




47 


529513 


586 


973580 


76 


555933 


661 


444067 


13 




48 


5298;'4 


585 


973535 


76 


556329 


060 


443671 


12 




49 


530215 


585 


973489 


76 


556725 


660 


443275 


11 




50 


530565 


584 


973444 


76 


557121 


659 


442879 


10 




51 


9.530915 


584 


9.973398 


76 


9.557517 


659 


10.442483 


9 




5fi 


531265 


583 


973352 


76 


557913 


659 


442087 


8 




53 


531614 


582 


973307 


76 


558308 


C58 


441692 


7 




54 


531963 


582 


973261 


76 


558702 


658 


441298 


6 




55 


532312 


581 


973215 


76 


559097 


657 


440903 


5 




56 


532661 


581 


973169 


76 


559491 


657 


440509 


4 




57 


533009 


580 


973124 


78 


559885 


656 


440115 


3 




58 


533357 


580 


973078 


76 


560279 


656 


439721 


2 




59 


533704 


579 


973032 


77 


560673 


C55 


439327 


1 




60 


534052 


578 


972986 


77 


561066 


C55 


438934 









Cosine 




Sine 


1 


Cotang. 




1 Tang. 


1 M. 





70 Degrees. 



64 



(20 Degrees.) a table op logarithmic 



M. 


1 Sine 


1 D. 


1 Cosine 


1 D- 


1 Tang. 


1 D- 


1 Cotang. 


1 







9.534052 


578 


9.972986 


77 


9.5)1066 


655 


10.4^89 !4 


60 




1 


534399 


577 


972940 


77 


531459 


654 


438541 


59 




2 


534745 


577 


972894 


77 


56 J 851 


654 


438149 


58 




3 


535092 


577 


972848 


77 


562244 


653 


437756 


57 




4 


535433 


576 


972802 


77 


562636 


653 


437364 


56 




5 


535783 


576 


972755 


77 


563028 


053 


436972 


55 




6 


536129 


575 


972709 


77 


533419 


652 


435581 


54 




7 


536474 


574 


972663 


77 


563811 


652 


436189 


53 




8 


536818 


574 


972617 


77 


564232 


651 


435798 


52 




9 


537163 


573 


972570 


77 


554592 


651 


435408 


51 




10 


537537 


573 


972524 


77 


554933 


650 


435017 


50 




11 


9.537851 


572 


9.972478 


77 


9.5653:3 


650 


10.434627 


49 




12 


538194 


572 


972431 


78 


56)763 


619 


434-237 


48 




13 


838538 


571 


972335 


78 


566153 


649 


433847 


47 




14 


538889 


571 


972338 


73 


5J6342 


649 


433458 


46 




15 


539223 


570 


972291 


78 


566932 


648 


433068 


45 




IG 


539535 


570 


972245 


78 


567320 


648 


432630 


44 




17 


539937 


569 


972198 


78 


567709 


647 


432291 


43 




18 


540249 


569 


972151 


78 


563093 


647 


431902 


42 




19 


5405^3 


563 


972105 


78 


563486 


646 


431514 


41 




20 


540931 


568 


972058 


78 


563373 


646 


431127 


40 




21 


9.541272 


567 


9.972011 


78 


9.539231 


645 


10.430739 


39 




22 


541613 


567 


971934 


73 


569548 


645 


430352 


38 




23 


541953 


566 


971917 


78 


570335 


645 


42991)5 


37 




24 


542293 


566 


971870 


78 


570422 


644 


429578 


36 




25 


542632 


565 


971823 


/8 


570309 


644 


429191 


35 




26 


542971 


565 


971776 


78 


571195 


643 


428835 


34 




27 


543310 


554 


971729 


79 


571581 


643 


428419 


33 




28 


543649 


564 


971682 


79 


571967 


642 


428033 


32 




29 


543987 


563 


971635 


79 


572352 


642 


427648 


31 




30 


544325 


563 


971588 


79 


572738 


642 


427262 


30 




31 


9.. 544663 


5G2 


9.971540 


79 


9.573123 


641 


10 426877 


29 




32 


545030 


532 


971493 


79 


573537 


641 


426493 


28 




33 


545338 


561 


971446 


79 


573832 


643 


426108 


27 




34 


545874 


561 


971398 


79 


574276 


640 


425724 


26 




35 


546011 


560 


971351 


79 


574660 


639 


425340 


25 




36 


546347 


560 


971333 


79 


575044 


639 


424956 


24 




37 


546683 


559 


971256 


79 


575427 


639 


424573 


23 




38 


547019 


559 


971208 


79 


575810 


638 


424193 


22 




39 


547354 


558 


971161 


79 


575193 


638 


423807 


21 




40 


' 547689 


558 


971113 


79 


576576 


637 


423424 


20 




41 


9.548324 


557 


9.971066 


80 


9.570958 


637 


10 423041 


19 




42 


548359 


557 


971018 


83 


577341 


636 


422659 


18 




43 


548693 


556 


970970 


83 


577723 


636 


4-22277 


17 




44 


549027 


556 


970922 


83 


578101 


636 


421896 


16 




45 


549360 


555 


970874 


83 


578486 


635 


421514 


15 




46 


549693 


555 


970827 


83 


578857 


635 


421133 


14 




47 


550025 


554 


970779 


83 


579243 


034 


4-20752 


13 




48 


550359 


554 


970731 


80 


579323 


634 


420371 


12 




49 


550692 


553 


970683 


83 


580309 


634 


419991 


11 




50 


551024 


553 


970335 


80 


580389 


633 


419611 


10 




51 


9.551356 


552 


9.970586 


83 


9.580769 


633 


10 419231 


9 




52 


551687 


552 


970538 


80 


581149 


632 


418851 


8 




53 


552018 


552 


970490 


83 


581528 


632 


418472 


7 




54 


552349 


551 


970442 


80 


581907 


632 


418093 


6 




55 


552680 


551 


970394 


83 


.58223() 


631 


417714 


5 




56 


553010 


550 


970345 


81 


5325()5 


631 


417335 


4 




57 


553341 


550 


970297 


81 


.583043 


630 


416957 


3 




58 


553670 


549 


970249 


81 


583422 


630 


416578 


o 




59 


554000 


549 


9702(30 


81 


.583800 


629 


416-200 


1 




60 


554329 


548 


970152 


81 


584177 


629 


4158-23 


1 




1 


Cosine 


1 


Sine 1 


I 


Cotang. 1 


1 


Tang. 1 


M. 1 










69 


Degr 


ties. 















SINES 


AND TANGENTS. (21 Degrees.] 


1 


65 


M. 


1 Sine 


1 D. 


1 Cosine 


D. 


Tang. 


1 D- 


Cotang. 1 


"60~ 





9.554329 


548 


9.970152 


81 


9.584177 


629 


10.415823 


1 


554658 


548 


970103 


81 


584555 


629 


415445 


59 


2 


554987 


547 


970055 


81 


584932 


628 


415068 


58 


3 


555315 


547 


970006 


81 


585309 


628 


414691 


57 


4 


555643 


546 


989957 


81 


585586 


627 


414314 


56 


5 


555971 


546 


969909 


81 


586082 


627 


413938 


55 


6 


556299 


545 


989860 


81 


585439 


627 


413561 


54 


7 


556626 


545 


989811 


81 


586815 


628 


413185 


53 


8 


556953 


544 


989762 


81 


587190 


623 


41-2810 


52 





557280 


544 


969714 


81 


587566 


625 


412434 


51 


10 


557606 


543 


969685 


81 


587941 


625 


412059 


50 


11 


9 557932 


543 


9.969618 


82 


9.588316 


625 


10.411684 


49 


12 


558258 


543 


969587 


82 


588891 


624 


411309 


48 


13 


558583 


542 


969518 


82 


589086 


624 


410934 


47 


14 


558909 


542 


989469 


82 


589440 


623 


410580 


46 


15 


559234 


541 


989420 


82 


589814 


623 


410186 


45 


16 


559558 


541 


969370 


82 


590188 


623 


409812 


44 


17 


559883 


540 


969321 


82 


590562 


622 


409438 


43 


18 


560207 


540 


989272 


82 


590935 


622 


409065 


42 


19 


560531 


539 


969223 


82 


591308 


622 


408892 


41 


20 


560855 


539 


969173 


82 


591681 


621 


408319 


40 


21 


9.561178 


538 


9.969124 


82 


9.592054 


621 


10.407946 


39 


22 


561501 


538 


989375 


82 


592426 


620 


407574 


38 


23 


561824 


537 


989025 


82 


592798 


620 


407202 


37 


24 


562116 


537 


988976 


82 


593170 


619 


406829 


36 


25 


562468 


536 


968926 


83 


593.542 


619 


406458 


35 


25 


562790 


536 


968877 


83 


593914 


618 


406086 


34 


27 


563112 


536 


983827 


83 


594-285 


618 


405715 


33 


28 


563433 


535 


958777 


83 


594656 


638 


405344 


32 


29 


563755 


535 


988728 


83 


595027 


617 


404973 


31 


30 


564075 


534 


988678 


83 


595398 


617 


404602 


30 


31 


9.584398 


534 


9.958628 


83 


9.595768 


617 


10.404232 


29 


32 


564716 


533 


968578 


83 


596138 


616 


403862 


28 


33 


565036 


533 


968528 


83 


596508 


616 


403492 


27 


34 


565358 


532 


968479 


83 


595878 


616 


403122 


26 


35 


565676 


532 


968429 


83 


597247 


615 


402753 


25 


36 


565995 


531 


968379 


83 


597616 


615 


402384 


24 


37 


586314 


531 


968329 


83 


597985 


615 


402015 


23 


38 


566632 


531 


968278 


83 


598354 


614 


401046 


22 


39 


566951 


530 


988228 


84 


598722 


614 


40 1278 


21 


40 


587269 


530 


968178 


84 


599091 


613 


400909 


20 


41 


9.567587 


529 


9.968128 


84 


9.599459 


613 


10.400541 


19 


42 


567904 


529 


968078 


84 


599827 


613 


400173 


18 


43 


568222 


528 


968027 


84 


600194 


612 


399808 


17 


44 


568539 


528 


967977 


84 


600562 


612 


399438 


16 


45 


568856 


528 


967927 


84 


600929 


611 


399071 


15 


46 


569172 


527 


967876 


84 


601296 


611 


398704 


14 


47 


569488 


527 


987826 


84 


601662 


611 


398338 


13 


48 


569804 


526 


967775 


84 


602029 


610 


397971 


12 


49 


570120 


526 


967725 


84 


602395 


610 


397605 


11 


50 


570435 


525 


967674 


84 


602761 


610 


397239 


10 


51 


9.570751 


525 


9.967624 


84 


9.603127 


609 


10.396873 


9 


52 


571066 


524 


967573 


84 


603493 


609 


396507 


8 


53 


571380 


524 


967522 


85 


603858 


609 


396142 


7 


54 


571695 


523 


967471 


85 


604223 


608 


395777 


6 


55 


572009 


523 


967421 


85 


604588 


608 


395412 


5 


56 


572323 


523 


967370 


85 


604953 


607 


395047 


4 


57 


572636 


522 


967319 


85 


605317 


607 


394683 


3 


58 


572950 


522 


967268 


85 


605682 


607 


394318 


2 


59 


5732G3 


521 


987217 


85 


606046 


606 


393954 


1 


60 


573575 


52J 


967166 


85 


606410 


606 


393590 


1 


"""7 


Cosine | 


1 


Sine 1 


1 


Cotang. 




Tang. 


M. 



68 Degrees. 
ellwood's test prob. — 5. 



66 



(23 Degrees.) A table of logarithmic 



1 M.l 




Sine 1 


D. 1 


Cosine | 


D- ( 


Tang. 1 


D 1 


Cotang 1 




9.573575 


521 


9.967166 


85 


9.606410 


606 


10.393590 


60 


' 1 


573888 


520 


967115 


85 


606773 


606 


393227 


59 


2 


574200 


520 


967064 


85 


607137 


605 


392863 


58 


3 


574512 


519 


907013 


85 


607500 


605 


392500 


57 


4 


574824 


519 


9G6961 


85 


607863 


604 


392137 


56 


1 5 


575136 


519 


960910 


85 


608225 


604 


391775 


55 


6 


575447 


518 


9GG859 


85 


608588 


604 


391412 


54 


7 


575758 


518 


9668U8 


85 


608950 


603 


.391050 


53 


8 


576009 


517 


906756 


80 


G09312 


603 


390688 


52 


9 


576379 


517 


906705 


86 


609674 


603 


390326 


51 


iO 


576689 


516 


966653 


80 


610036 


602 


389964 


50 


11 


9.576999 


516 


9.966602 


80 


9.610397 


602 


10.389603 


49 


12 


5773U9 


516 


906550 


86 


610759 


602 


389241 


48 


13 


577618 


515 


966499 


86 


011120 


601 


388880 


47 


14 


577927 


.515 


966447 


86 


611480 


601 


388520 


46 


15 


578236 


514 


906395 


80 


G11841 


601 


388159 


45 


16 


578545 


514 


900344 


86 


612201 


600 


387799 


44 


17 


578853 


513 


900292 


86 


612501 


600 


387439 


43 


18 


579162 


513 


906240 


86 


612921 


600 


387079 


42 


19 


579470 


513 


900188 


86 


613281 


599 


386719 


41 


20 


579777 


512 


900136 


86 


613G41 


599 


380359 


40 


21 


9.580085 


512 


9.966085 


87 


9.614000 


598 


10.386000 


39 


22 


580392 


511 


966033 


87 


614359 


598 


385641 


38 


23 


580699 


511 


965981 


87 


614718 


598 


385282 


37 


24 


581005 


.511 


905928 


87 


615077 


597 


384923 


36 


25 


581312 


510 


905876 


87 


615435 


597 


384505 


35 


26 


581618 


510 


905824 


87 


615793 


597 


384207 


34 


27 


581924 


509 


965772 


87 


616151 


590 


383849 


3:) 


28 


582229 


509 


965720 


87 


616509 


596 


383491 


32 


29 


582535 


509 


965668 


87 


616867 


596 


383133 


31 


30 


582840 


508 


905615 


87 


617224 


595 


382776 


30 


31 


9.583145 


508 


9.905563 


87 


9.617582 


595 


10.382418 


29 


32 


583449 


507 


965511 


87 


617939 


595 


382061 


28 


33 


583754 


507 


905458 


87 


618295 


594 


381705 


27 


34 


584058 


506 


905406 


87 


618652 


594 


381.348 


26 


35 


584361 


506 


965353 


8H 


619008 


594 


380992 


25 


36 


584665 


5U6 


905301 


88 


619364 


593 


380636 


24 


37 


584908 


505 


965248 


88 


619721 


593 


380279 


23 


38 


585272 


505 


965195 


88 


620076 


593 


379924 


22 


39 


585574 


504 


965143 


88 


620432 


592 


379568 


21 


40 


585877 


504 


965090 


88 


620787 


592 


379213 


20 


41 


9.586179 


503 


9.965037 


88 


9.621142 


592 


10.378858 


19 


42 


586482 


503 


964984 


83 


621497 


591 


378503 


18 


43 


586783 


503 


964931 


88 


621852 


591 


378148 


17 


44 


587U85 


502 


964879 


88 


622207 


590 


377793 


16 


45 


.587386 


502 


904826 


88 


622561 


590 


377439 


15 


46 


587688 


501 


904773 


88 


622915 


590 


377085 


14 


47 


587989 


501 


904719 


88 


6232G9 


589 


37673] 


13 


48 


588289 


501 


904666 


89 


623623 


589 


376.377 


12 


49 


588590 


500 


964613 


89 


623976 


589 


376024 


11 


50 


588890 


500 


964560 


89 


624330 


588 


375670 


10 


51 


9.589190 


499 


9.964507 


89 


9.624683 


588 


10.375317 


9 


52 


589489 


499 


904454 


89 


625036 


588 


374964 


8 


53 


589789 


499 


964400 


89 


625388 


587 


374612 


7 


54 


590088 


498 


964347 


89 


625741 


587 


374259 


6 


55 


590387 


498 


964294 


89 


626093 


587 


373907 


5 


56 


590686 


497 


964240 


89 


62(>445 


586 


373555 


4 


57 


590984 


497 


964187 


89 


62()797 


586 


373203 


3 


58 


591282 


497 


904133 


89 


627149 


586 


372851 


o 


59 


591580 


496 


9<)4080 


89 


627501 


585 


372499 


1 


' 60 

1 


591878 


496 


964026 


89 


627852 


585 


372148 


ri 


j 


1 Cosine 




i Sine 


1 


1 Cobang. 


1 


1 Tang. 








67 


Degr 


ees. 






MBnaatf 







SINES 


AND TANGE^fTS. (23 Degrees.) 




67 


1 M. 


Sine 


D. 


Cosine 


D. 


Tang. 


1 ». 


Cotang. 







9.591878 


496 


9.964026 


89 


9.627852 


585 


10.372148 


60 


1 


592176 


495 


963972 


89 


628203 


585 


371797 


59 


2 


592473 


495 


953919 


89 


628554 


585 


37144G 


58 


3 


592770 


495 


963865 


90 


628905 


584 


371095 


57 


4 


593067 


494 


9;33811 


90 


629255 


534 


370745 


56 


5 


593363 


494 


953757 


90 


629506 


583 


370394 


55 


6 


593659 


493 


953704 


90 


629956 


583 


370044 


54 


7 


593955 


493 


963650 


90 


630306 


583 


369694 


53 


8 


594251 


493 


963596 


90 


630656 


583 


369344 


52 


9 


594547 


492 


953542 


90 


631005 


582 


368995 


51 


10 


594842 


492 


963488 


90 


631355 


582 


368645 


50 


11 


9.595137 


491 


9.963434 


90 


9.631704 


582 


10.368296 


49 


12 


595432 


491 


963379 


93 


632353 


581 


367947 


48 


13 


595727 


491 


963325 


90 


632401 


581 


367599 


47 


14 


596021 


490 


963271 


90 


632750 


581 


367250 


46 


15 


596315 


490 


963217 


90 


633098 


580 


366902 


45 


16 


596609 


489 


963163 


9J 


633447 


580 


365553 


44 


17 


596903 


489 


963108 


91 


633795 


580 


356205 


43 


18 


597193 


489 


963054 


91 


634143 


579 


365857 


42 


19 


597490 


488 


962999 


91 


634490 


579 


355510 


41 


20 


597783 


488 


962945 


91 


634838 


579 


305162 


40 


21 


9.598075 


487 


9.962890 


91 


9.635185 


578 


10.364815 


39 


22 


598368 


487 


962836 


91 


635532 


578 


364468 


38 


23 


598660 


487 


962781 


91 


635379 


578 


364121 


37 


24 


598952 


486 


962727 


91 


636223 


577 


353774 


36 


25 


599244 


486 


962672 


91 


636572 


577 


363423 


35 


26 


599536 


485 


962617 


91 


636919 


577 


363081 


34 


27 


599827 


485 


962562 


91 


637265 


577 


362735 


33 


28 


600118 


485 


962508 


91 


637611 


570 


332389 


32 


29 


600409 


484 


952453 


91 


637956 


576 


362044 


31 


30 


600700 


484 


962398 


92 


638302 


576 


361698 


30 


31 


9.600990 


484 


9.962343 


92 


9.638647 


575 


10.331353 


29 


32 


60128:) 


483 


962288 


92 


638992 


575 


361008 


28 


33 


601570 


483 


962233 


92 


639337 


575 


360663 


27 


34 


6018(30 


482 


962178 


92 


639682 


574 


360318 


26 


35 


602150 


482 


962123 


92 


640027 


574 


359973 


25 


36 


602439 


482 


962067 


92 


640371 


574 


359529 


24 


37 


602728 


481 


962012 


92 


640716 


573 


359284 


23 


38 


603017 


481 


961957 


92 


641050 


573 


358940 


22 


39 


603305 


481 


931902 


92 


641404 


573 


353596 


21 


40 


603594 


480 


961846 


92 


641747 


572 


358253 


20 


41 


9.603882 


480 


9.961791 


92 


9.642091 


572 


10.357909 


19 


42 


604170 


479 


951735 


92 


642434 


572 


357566 


18 


43 


004457 


479 


961680 


92 


612777 


5T2 


357223 


17 


44 


604745 


479 


961624 


93 


643120 


571 


356880 


16 


45 


605032 


478 


961569 


93 


643463 


571 


355537 


15 


46 


605319 


478 


961513 


93 


643836 


571 


355194 


14 


47 


605605 


478 


961458 


93 


644148 


570 


355852 


13 


48 


605892 


477 


961402 


93 


644490 


570 


355510 


12 


49 


606179 


477 


961346 


93 


644832 


570 


355168 


11 


50 


606465 


476 


961-290 


93 


645174 


569 


354826 


10 


51 


9.606751 


476 


9.961235 


93 


9.645516 


569 


10.354484 


9 


52 


607035 


476 


981179 


93 


645857 


569 


354143 


8 


53 


607322 


475 


931 123 


93 


646199 


569 


353801 


7 


54 


607007 


475 


931067 


93 


646540 


568 


353460 


6 


, 55 


607892 


474 


961011 


93 


646881 


558 


353119 


5 


56 


608177 


474 


930955 


93 


647222 


568 


352778 


4 


57 


608461 


474 


960899 


93 


647552 


567 


352438 


3 


58 


608745 


473 


960843 


94 


647903 


567 


352097 


2 


59 


609029 


473 


960786 


94 


648243 


557 


351757 


1 


60 


609313 


473 


960730 


94 


648583 


566 


351417 





1 


(>osine 




Sine 1 




Cotang. 




Tang. 


M. 



66 Degrees. 



68 




[24 Degrees.) a 


TABLE OP LOGARITHMIC 




M 


1 Sine 


1 D 


1 Cosine 


i D. 


1 Tang. 


! D- 


1 Cotang. 


1 





9.609313 


473 


9.960730 


94 


9.648583 


566 


10.351417 


60 


1 


609597 


472 


960674 


94 


648923 


566 


351077 


59 


2 


609880 


472 


960618 


94 


6492(53 


566 


350737 


58 


3 


610164 


472 


9fi0561 


94 


649602 


56G 


3.50398 


57 


4 


610447 


471 


960.505 


94 


649942 


565 


350058 


56 


5 


610729 


471 


960448 


94 


650281 


565 


3497 19 


55 


6 


611012 


470 


9603J2 


94 


650620 


565 


249380 


54 


7 


6J 1-294 


470 


960335 


94 


6.50959 


564 


349041 


53 


8 


611576 


470 


960270 


94 


651297 


564 


648703 


52 


9 


611858 


469 


960222 


94 


651636 


5(54 


348364 


51 


10 


612J40 


469 


960165 


94 


651974 


563 


348026 


50 


11 


9.612421 


469 


9.960109 


95 


9.652312 


563 


10.347688 


49 


12 


612702 


468 


960052 


95 


652650 


563 


347350 


48 


13 


612983 


468 


959995 


i)5 


m2988 


563 


347012 


47 


14 


613264 


467 


959938 


95 


C53326 


562 


346674 


46 


15 


613545 


467 


950882 


95 


653663 


562 


346337 


45 


16 


613825 


467 


950825 


95 


G54000 


562 


34(5000 


44 


17 


614105 


4()6 


959768 


95 


654337 


561 


345663 


43 


18 


614385 


466 


9.59711 


95 


654674 


561 


345326 


42 


19 


614G65 


466 


959654 


95 


65.5011 


561 


344989 


41 


20 


614944 


465 


959596 


95 


655348 


561 


344652 


40 


21 


9.615223 


465 


9.9,595.39 


95 


9.655684 


560 


10.344316 


39 


22 


615502 


465 


959482 


95 


656020 


560 


343980 


38 


23 


615781 


464 


959425 


95 


656356 


560 


343(544 


37 


24 


616000 


464 


959368 


95 


656692 


5.59 


343308 


36 


25 


616338 


464 


959310 


96 


657028 


5.59 


342972 


35 


26 


OlfiOlG 


463 


959253 


96 


657364 


559 


342(53(> 


34 


27 


616894 


463 


9.59195 


96 


657699 


559 


342301 


33 


28 


617172 


462 


9.59138 


96 


6.58034 


558 


341966 


32 


21) 


617450 


462 


959081 


96 


058369 


558 


341(^31 


31 


30 


617727 


462 


959023 


93 


6.58704 


558 


341296 


30 


31 


9.618004 


461 


9.958965 


96 


9.659039 


5.58 


10.340961 


29 


32 


618281 


461 


958908 


96 


659373 


557 


340627 


28 


33 


618558 


461 


958850 


96 


659708 


557 


340292 


27 


34 


618834 


460 


9.58792 


96 


660042 


557 


3399.58 


26 


35 


019110 


460 


958734 


96 


660376 


557 


339(524 


25 


36 


01 9:186 


460 


958677 


96 


660710 


5.56 


3392m) 


24 


37 


619G62 


459 


9.58619 


96 


661043 


556 


338957 


23 


38 


61U938 


459 


958561 


96 


661377 


556 


3386-2:^ 


22 


39 


620213 


4.59 


958503 


97 


661710 


5.55 


3.38290 


21 


40 


620488 


458 


958445 


97 


662043 


555 


337957 


20 


41 


9.620763 


458 


9.958387 


97 


9.662376 


555 


10.337624 


19 


42 


621038 


457 


958329 


97 


662709 


5.54 


337291 


18 


43 


621313 


4.57 


958271 


97 


6(53042 


.554 


33(59.58 


17 


44 


621.587 


457 


9.58213 


97 


6(5.3375 


551 


336(525 


16 


45 


621861 


4.56 


9.58154 


97 


6637(;7 


554 


33(5293 


15 


46 


622135 


456 


958096 


97 


(5(5403!) 


553 


3.359(51 


14 


47 


622409 


456 


9.58038 


97 


664371 


553 


335(529 


13 


48 


622682 


455 


957979 


97 


664703 


553 


335297 


12 


49 


622956 


4.55 


9.57921 


97 


6(55035 


553 


33-19(55 


11 


50 


623229 


455 


9578(53 


97 


665366 


552 


334634 


10 


51 


9.623.502 


454 


9.957804 


97 


9.66.5(597 


552 


10.3.3-1303 


9 


52 


623774 


4.54 


9577-16 


98 


6()(50e9 


552 


333971 


8 


53 


624047 


4.54 


957687 


98 


666.360 


551 


333(510 


7 


54 


624319 


453 


957()28 


98 


666(591 


551 


3.33309 


6 


55 


624501 


453 


957570 


98 


6()7021 


551 


33>979 


5 


56 


6248f;3 


453 


9.57511 


98 


6673.52 


551 


.332(548 


4 


57 


625135 


4.52 


957452 


98 


6(w(582 


5.50 


332318 


3 


58 


625406 


452 


957393 


98 


6(58013 


5.50 


331987 


2 


59 


625(i77 


4.52 


957335 


98 


6(58343 


.550 


331(5.57 


1 


60 


625948 


451 


957276 


98 


6(^8(572 


550 


331.328 





1 


Cosine | 


1 


Sine 1 


! 


Cotang. 1 


1 


Tang. 1 


M. 



65 Degrees. 







SINES AND TANGENTS. 


(25 Degrees.) 




6< 


) 


M. 


1 Sine 


D. 


Cosine 


D. 


Tang. 


D. 


1 Cotang. 









9.625948 


451 


9.957276 


98 


9.668873 


550 


10.331327 


60 




1 


626219 


451 


957217 


98 


689002 


549 


330998 


59 




2 


626490 


451 


957158 


98 


689332 


549 


330668 


58 




3 


62G760 


450 


957099 


98 


669861 


549 


330339 


57 




4 


627030 


450 


957040 


98 


669991 


548 


330009 


58 




5 


627300 


450 


956981 


98 


670320 


548 


329r^80 


55 




6 


627570 


449 


956921 


99 


67U649 


548 


329351 


54 




7 


627840 


449 


956862 


99 


670977 


548 


329023 


53 




8 


628109 


449 


956803 


99 


671306 


547 


328894 


52 




9 


628378 


448 


956744 


99 


671634 


547 


328366 


51 




10 


628647 


448 


956684 


99 


671963 


547 


328037 


50 




11 


9.628916 


447 


9.956625 


99 


9.672291 


547 


10.327709 


49 




12 


629185 


447 


956566 


99 


672619 


546 


327381 


48 




13 


629453 


447 


956508 


99 


672947 


546 


327053 


47 




14 


629721 


446 


956447 


99 


673274 


545 


326726 


46 




15 


629989 


446 


956387 


99 


673802 


546 


326398 


45 




16 


630257 


446 


956327 


99 


673929 


545 


328071 


44 




17 


630524 


446 


956268 


99 


674257 


545 


325743 


43 




18 


630792 


445 


958208 


100 


674584 


545 


32.5416 


42 




19 


631059 


445 


956148 


100 


674910 


544 


325090 


41 




20 


631326 


445 


956089 


100 


675237 


544 


324763 


40 




21 


9.631593 


444 


9.956029 


100 


9.675564 


544 


10.324436 


39 




22 


031859 


444 


955939 


100 


675890 


544 


324110 


38 




23 


632125 


444 


955909 


100 


676216 


543 


323784 


37 




24 


632392 


443 


955849 


100 


676543 


543 


323457 


36 




25 


632658 


443 


955789 


100 


676869 


543 


323131 


35 




26 


632923 


443 


955729 


100 


677194 


543 


322806 ■ 


34 




27 


633189 


442 


955689 


100 


677520 


542 


322480 


33 




28 


633454 


442 


955609 


100 


677846 


542 


322154 


32 




29 


633719 


442 


955548 


100 


678171 


542 


321829 


31 




30 


633984 


441 


955488 


100 


67d496 


542 


321504 


30 




31 


9.634249 


441 


9.955428 


101 


9.678821 


541 


10.321179 


29 




32 


634514 


440 


9553e« 


101 


679 J 46 


541 


320854 


28 




33 


634778 


440 


955307 


101 


679471 


541 


323529 


27 




34 


635042 


440 


955247 


301 


679795 


541 


320205 


26 




35 


635306 


439 


955186 


101 


680120 


540 


319880 


25 




36 


635570 


439 


955126 


101 


680444 


540 


319556 


24 




37 


635834 


439 


955005 


101 


680768 


540 


319232 


23 




38 


636097 


438 


955005 


101 


681092 


540 


318908 


22 




39 


630360 


438 


954944 


101 


681416 


539 


318.584 


21 




40 


636623 


438 


954883 


101 


681740 


539 


318280 


20 




41 


9.636886 


437 


9.954823 


101 


9.682063 


539 


10.317937 


19 




42 


637148 


437 


954762 


101 


082387 


539 


317613 


18 




43 


637411 


437 


954701 


101 


682710 


538 


317290 


17 




44 


637673 


437 


954640 


101 


683033 


538 


318967 


16 1 




45 


637935 


436 


954579 


101 


683358 


538 


316844 


15 




46 


638197 


436 


954518 


102 


683679 


538 


318321 


14 




47 


638458 


436 


954457 


102 


684001 


537 


315999 


13 




48 


638720 


435 


954396 


102 


684324 


537 


315676 


12 




49 


638981 


435 


954335 


102 


684648 


537 


315354 


11 




50 


639242 


435 


954274 


102 


684968 


537 


315032 


10 




51 


9.639503 


434 


9.954213 


102 


9.685290 


536 


10.314710 


9 




52 


639764 


434 


954152 


102 


685672 


536 


314388 


8 




53 


640024 


434 


954090 


102 


685934 


536 


314086 


7 




54 


640284 


433 


954029 


102 


686255 


536 


313745 


6 




55 


640544 


433 


953968 


102 


686577 


535 


313423 


5 




56 


640804 


433 


953906 


102 


686898 


535 


313102 


4 




57 


641004 


432 


953845 


102 


687219 


535 


312781 


3 




58 


641324 


432 


953783 


]02 


687540 


535 


312460 


2 




59 


641584 


432 


953722 


103 


687861 


534 


312139 


1 




1 60 


641842 


431 


953660 


103 


688182 


534 


311818 







[ 


Cosine 


1 


Sine 


1 


Cotang. 




Tang. 


M. 





64 Degrees. 



70 


< 


[26 Degrees.) a 


TABLE or LOGARITHMIC 




M 


Sine 


1 D- 


Cosine 


D. 


Tang. 


D. 


Cotang. 







9.041842 


431 


9.953000 


103 


9.688182 


534 


10.311818 


60 


1 


042101 


431 


953599 


103 


688502 


534 


311498 


59 


2 


C423G0 


431 


953537 


103 


688823 


534 


311177 


58 


3 


642018 


430 


953475 


103 


689143 


533 


310857 


57 


4 


642877 


430 


953413 


103 


689463 


533 


310537 


56 


5 


643135 


430 


953352 


103 


689783 


533 


310217 


55 


6 


643393 


430 


953290 


103 


690103 


533 


309897 


54 


7 


643G50 


429 


953228 


103 


690423 


533 


309577 


53 


8 


643908 


429 


953166 


103 


690742 


532 


309258 


52 


9 


644165 


429 


953104 


103 


691062 


532 


308938 


51 


10 


644423 


428 


953042 


103 


691381 


532 


308619 


50 


11 


9.644680 


428 


9 952980 


104 


9.691700 


531 


10.308300 


49 


12 


044930 


428 


952918 


104 


692019 


531 


307981 


48 


13 


645193 


427 


952855 


104 


692338 


531 


307662 


47 


14 


645450 


427 


952793 


104 


692656 


531 


307344 


46 


15 


645706 


427 


952731 


104 


692975 


531 


307025 


45 


16 


645962 


426 


952669 


104 


693293 


530 


306707 


44 


17 


646218 


426 


952606 


104 


693612 


530 


306388 


43 


18 


646474 


426 


952544 


104 


693930 


530 


306070 


42 


19 


646729 


425 


952481 


104 


694248 


.530 


305752 


41 


20 


646984 


425 


952419 


104 


694566 


529 


305434 


40 


21 


9.647240 


425 


9.952356 


104 


9.694883 


529 


10.305117 


39 


22 


647494 


424 


952294 


104 


695201 


529 


304799 


38 


23 


647749 


424 


952231 


104 


695518 


529 


304482 


37 


24 


648004 


424 


9521C8 


105 


695836 


529 


304164 


36 


25 


648258 


424 


952106 


105 


690153 


528 


303847 


35 


26 


648512 


423 


9^043 


105 


696470 


528 


303530 


34 


27 


648766 


423 


951980 


105 


690787 


528 


303213 


33 


28 


649020 


423 


951917 


105 


697103 


528 


302897 


32 


29 


649274 


422 


951854 


105 


697420 


527 


30-2580 


31 


30 


649527 


422 


951791 


105 


697736 


527 


302264 


30 


31 


9.649781 


422 


9.951728 


105 


9.698053 


527 


10.301947 


29 


32 


650034 


422 


951665 


105 


693369 


527 


301631 


28 


33 


650287 


421 


951602 


105 


698685 


526 


301315 


27 


34 


650539 


421 


951539 


105 


699001 


526 


300999 


26 


35 


650792 


421 


951476 


105 


699316 


526 


300684 


25 


36 


651044 


420 


951412 


105 


699632 


526 


300368 


24 


37 


651297 


420 


951349 


106 


699947 


526 


300053 


23 


38 


651549 


420 


951280 


106 


700203 


525 


299737 


22 


39 


651800 


419 


951222 


106 


700578 


525 


299422 


21 


40 


652052 


419 


951159 


106 


700893 


525 


299107 


20 


41 


9.652304 


419 


9.951096 


106 


9.701208 


524 


10.298792 


19 


42 


652555 


418 


951032 


106 


701523 


524 


298477 


18 


43 


652806 


418 


950908 


100 


701837 


5-24 


298163 


17 


44 


653057 


418 


950905 


106 


702152 


524 


297848 


16 


45 


653308 


418 


950841 


106 


702406 


524 


297534 


15 


46 


653558 


417 


950778 


106 


702780 


523 


297220 


14 


47 


653808 


417 


950714 


106 


703095 


523 


296905 


13 


48 


654059 


417 


950650 


106 


703409 


523 


29(m91 


12 


49 


654309 


416 


950586 


106 


703723 


523 


29t)277 


11 


50 


6545.58 


416 


950522 


107 


704036 


522 


295964 


10 


5] 


9.654808 


416 


9.950458 


107 


9.704350 


522 


10.295050 


9 


52 


655058 


416 


950394 


107 


704663 


5-22 


295337 


8 


53 


655307 


415 


950330 


107 


704977 


5-22 


2950-2:1 


7 


54 


655556 


415 


95026() 


107 


705290 


522 


294710 


6 


55 


655805 


415 


950202 


107 


705603 


521 


294397 


5 


5(5 


650054 


414 


950138 


107 


705916 


521 


294084 


4 


57 


656302 


414 


950074 


107 


70(>228 


521 


293772 


3 


58 


656551 


414 


950010 


107 


706541 


521 


293459 


o 


59 


650799 


413 


949945 


107 


706854 


521 


293140 


1 


60 


657047 


413 


949881 


107 


707166 


520 


29-2834 







CoHine 




Sine 




Cotang. 


1 


Tang. 


|M_ 



63 Degrees. 







SINES AND TANGENTS. (27 DcgrCCS.) 




7: 


L 


M. 


Sine 


1 D. 


Cosine 


1 D. 
107 


1 Tang. 
9.707166 


1 »• 
520 


Cotang. 
10.292834 


1 

60 







9.657047 


413 


9.949881 




1 


657295 


413 


949816 


107 


707478 


520 


292522 


59 




2 


657542 


412 


9497.52 


107 


707790 


520 


292210 


58 




3 


657790 


412 


949688 


108 


708102 


520 


291898 


57 




4 


658037 


412 


919323 


108 


708414 


519 


291586 


56 




5 


658284 


412 


949558 


108 


708726 


519 


291274 


55 




6 


658531 


411 


949494 


108 


709037 


519 


29J933 


54 




7 


658778 


411 


949429 


108 


709349 


519 


290851 


53 




8 


659025 


411 


949384 


108 


709580 


519 


290340 


52 




9 


659271 


410 


949300 


108 


709971 


518 


290029 


51 




10 


659517 


410 


949235 


108 


710282 


518 


289718 


50 




11 


9.659763 


410 


9.949170 


108 


9.710593 


518 


10.239407 


49 




12 


660009 


409 


949105 


108 


710904 


518 


289096 


48 




13 


660255 


409 


949040 


108 


711215 


818 


288735 


47 




14 


660501 


409 


948975 


108 


711525 


517 


288475 


46 




15 


660746 


403 


948910 


108 


711836 


517 


283164 


45 




16 


660991 


408 


948845 


108 


712146 


517 


287854 


44 




17 


661235 


408 


948780 


109 


712453 


517 


287.544 


43 




18 


661481 


408 


948715 


109 


712766 


516 


287234 


42 




19 


661726 


407 


943':)50 


10.) 


713076 


516 


288924 


41 




20 


661970 


407 


948584 


109 


713388 


516 


286614 


40 




21 


9.662214 


407 


9.948519 


109 


9.713693 


516 


10.288304 


39 




22 


662459 


407 


948454 


109 


714005 


516 


285905 


38 




23 


662703 


408 


948388 


109 


714314 


515 


285888 


37 




24 


662946 


406 


948323 


109 


714624 


515 


285376 


36 




25 


663190 


408 


948257 


100 


714933 


515 


285067 


35 




25 


663433 


405 


948192 


109 


715242 


515 


284758 . 


34 




27 


663677 


405 


943123 


109 


715551 


514 


284449 


33 




23 


663920 


405 


943080 


109 


7153G0 


514 


234140 


32 




29 


664163 


405 


947995 


110 


716168 


514 


283832 


31 




30 


664406 


404 


947929 


110 


716477 


514 


283523 


30 




31 


9.664048 


404 


9.947883 


110 


9.716785 


514 


10.283215 


29 




32 


664891 


404 


947797 


110 


717093 


513 


232937 


28 




33 


665133 


403 


947731 


110 


717401 


513 


232599 


27 




34 


685375 


403 


947865 


110 


717709 


513 


282291 


26 




35 


66.5017 


403 


947600 


110 


718017 


513 


281933 


25 




36 


665859 


402 


947533 


no 


718325 


513 


281675 


24 




37 


686103 


402 


947467 


no 


718633 


512 


281367 


23 




38 


666342 


402 


947401 


no 


718940 


512 


281080 


22 




39 


686583 


402 


947.335 


110 


719248 


512 


280752 


21 




40 


656824 


401 


947289 


110 


719555 


512 


280445 


20 




41 


9.687065 


401 


9.947203 


no 


9.719362 


512 


10.280138 


19 




42 


6C7305 


401 


947136 


111 


720169 


511 


279831 


18 




43 


667546 


401 


947070 


111 


720476 


511 


279524 


17 




44 


667786 


400 


947004 


111 


720783 


511 


279217 


16 




45 


668027 


400 


946937 


111 


721089 


511 


278911 


15 




48 


608267 


400 


946871 


111 


721398 


511 


278804 


14 




47 


668508 


399 


948804 


111 


721702 


510 


278293 


13 




48 


608748 


399 


948733 


111 


722009 


510 


277991 


12 




49 


668938 


399 


943671 


111 


722315 


510 


277685 


11 




50 


689225 


399 


946604 


111 


722621 


510 


277379 


10 




51 


9.6694G4 


398 


9.946538 


111 


9.722927 


510 


10.277073 


9 




52 


669703 


398 


946471 


111 


723232 


509 


276768 


8 




53 


669942 


398 


946404 


111 


723538 


509 


276462 


7 




54 


670181 


397 


948337 


111 


723844 


509 


276156 


6 




55 


670419 


397 


946270 


112 


724149 


509 


275851 


5 




53 


670658 


397 


946203 


112 


724454 


509 


275546 


4 




57 


670898 


397 


946136 


112 


724759 


508 


275241 


3 




58 


671134 


393 


946069 


112 


725065 


503 


274935 


2 




59 


671372 


398 


946002 


n2 


725369 


508 


274631 


1 




60 


671609 


396 


945935 


112 


725674 


508 


274326 







" 


Cosine 




Sine 




Cotang. 




Tang. 


M. 





62 Degrees. 



72 



(28 Degrees.) a tabt^e or logarithmtc 



M. 


1 Sine 


1 D. 


( Cosine 


1 D. 


1 Tang. 


I D. 


1 Cotang. 1 







9.671609 


396 


9.945935 


112 


9.72.5674 


508 


10.274326 


60 




1 


671847 


395 


94.5868 


112 


725979 


508 


274021 


59 




2 


672084 


395 


94.5800 


112 


7262o4 


507 


273716 


58 




3 


672321 


395 


945733 


112 


726588 


507 


273412 


57 




4 


672558 


395 


945666 


112 


726892 


507 


273108 


56 




5 


672795 


394 


945598 


112 


727197 


507 


272803 


55 




6 


673032 


394 


945531 


112 


727501 


507 


272499 


54 




7 


673268 


394 


945464 


113 


727805 


506 


272195 


53 




8 


673505 


394 


945396 


113 


728109 


506 


271891 


52 




9 


673741 


393 


945328 


113 


728412 


500 


27J588 


51 




10 


673977 


393 


945261 


113 


728716 


506 


271284 


50 




11 


9.674213 


393 


9.945193 


113 


9.729020 


506 


10.270980 


49 




12 


674448 


392 


945125 


113 


729323 


505 


270677 


48 




13 


674684 


392 


9450.58 


113 


729626 


505 


270374 


47 




14 


674919 


392 


944990 


113 


729929 


505 


270071 


46 




15 


675155 


392 


9449-22 


113 


730233 


505 


269767 


45 




16 


675390 


391 


944854 


113 


730535 


505 


269465 


44 




17 


675624 


391 


944786 


113 


730838 


504 


2(59162 


43 




18 


675859 


391 


944718 


113 


731141 


504 


268859 


42 




19 


676094 


391 


944650 


113 


731444 


504 


268556 


41 




20 


676328 


390 


944582 


114 


731746 


504 


268254 


40 




21 


9.676562 


390 


9.944514 


114 


9.732048 


504 


10.267952 


39 




22 


676796 


390 


944446 


114 


732351 


503 


267649 


38 




23 


677030 


390 


944377 


114 


732653 


503 


267347 


37 




24 


677264 


389 


944309 


114 


732955 


503 


267045 


36 




25 


677498 


389 


944241 


114 


733257 


503 


266743 


35 




26 


677731 


389 


944172 


114 


733558 


503 


266442 


34 




27 


677964 


388 


944104 


114 


733860 


502 


266140 


33 




2« 


678197 


388 


944036 


114 


734162 


502 


265838 


32 




29 


678430 


388 


9439G7 


114 


73440^ 


502 


265.537 


31 




30 


678G03 


388 


943899 


114 


734764 


502 


265236 


30 




31 


9.678895 


387 


9.943830 


114 


9.735066 


502 


10.264934 


29 




32 


679128 


387 


943761 


114 


735367 


502 


2f)4533 


28 




33 


679360 


387 


943093 


115 


735668 


501 


264332 


27 




34 


079592 


387 


943024 


115 


7359G9 


501 


264031 


26 




35 


679824 


386 


943555 


115 


736209 


501 


263731 


25 




36 


680056 


386 


943488 


115 


736570 


501 


2634.30 


24 




37 


680288 


386 


943417 


115 


736871 


501 


263129 


23 




38 


680519 


385 


943348 


115 


737171 


500 


2628-29 


22 




39 


680750 


385 


943279 


115 


737471 


500 


262529 


21 




40 


680982 


385 


943210 


115 


737771 


500 


262229 


20 




41 


9.681213 


385 


9.943141 


115 


9.738071 


500 


10.261929 


19 




42 


681443 


384 


943072 


115 


738371 


500 


261629 


18 




43 


681674 


384 


943003 


115 


738f)71 


499 


261329 


17 




44 


681905 


384 


942934 


115 


738971 


499 


261029 


10 




45 


682135 


384 


942864 


115 


739271 


499 


260729 


15 




46 


682365 


383 


942795 


110 


739570 


499 


200430 


14 




47 


682595 


383 


942726 


116 


739870 


499 


260130 


13 




48 


682825 


383 


942656 


116 


740109 


499 


259831 


12 




49 


683055 


383 


942587 


116 


740468 


498 


259532 


11 




50 


683284 


382 


942517 


116 


740767 


498 


259233 


10 




51 


9.683514 


382 


9.942448 


116 


9.741066 


498 


10.2.58034 


9 




52 


683743 


382 


942378 


116 


741365 


498 


2.58635 


8 




53 


683972 


382 


942308 


116 


741664 


498 


258336 


7 




54 


684201 


381 


942239 


116 


741962 


497 


258038 


6 




55 


684430 


381 


942169 


110 


742261 


497 


257739 


5 




56 


684658 


381 


942099 


116 


742559 


497 


257441 


4 




57 


684887 


:i80 


942029 


116 


7428.'58 


497 


257142 


3 




58 


685115 


:m 


9419.'')9 


116 


74315() 


497 


2;->6844 


2 




59 


685343 


380 


941889 


117 


7434.')4 


497 


256.546 


1 




60 


685571 


380 


941819 


117 


743752 


496 


25(5248 







J 


~C08iiie~'|' 


1 


Sine 1 


1 


Cotang. 1 


1 


Tang. 1 M. '| 





61 Degrees 







SINES AND TANGENTS. (29 Degrees.) 




73 


M. 


Sine 


D- 


Cosine 


D. 


1 Tang. 


D. 


Cotang. 


1 





9.685571 


380 


9.941819 


117 


9.743752 


496 


10.256248 


60 


1 


685799 


379 


941749 


117 


744050 


495 


255950 


59 


2 


6860-27 


379 


941679 


117 


744348 


495 


255652 


58 


3 


686254 


379 


941609 


117 


744045 


496 


255355 


57 


4 


686482 


379 


941539 


117 


744943 


495 


255057 


56 


5 


686709 


378 


941469 


117 


745240 


495 


254760 


55 


6 


686936 


378 


941398 


117 


745538 


495 


254462 


54 


7 


687163 


378 


941328 


117 


745335 


495 


254165 


53 


8 


687389 


378 


941258 


117 


746132 


495 


253868 


52 


9 


687616 


377 


941187 


117 


743429 


495 


25.3571 


51 


10 


687843 


377 


941117 


117 


740726 


495 


253274 


50 


11 


9.688069 


377 


9.941046 


118 


9.747023 


494 


10.252977 


49 


12 


688295 


377 


940975 


118 


747319 


494 


252681 


48 


13 


688521 


376 


940005 


118 


747616 


494 


252384 


47 


14 


688747 


376 


940834 


118 


747913 


494 


252087 


46 


15 


688972 


376 


940763 


118 


748209 


494 


251791 


45 


16 


689198 


376 


940593 


118 


748505 


493 


251495 


44 


17 


689423 


375 


940622 


118 


748801 


493 


251199 


43 


18 


689648 


375 


940551 


118 


749097 


493 


250903 


42 


19 


689373 


375 


940480 


118 


749393 


493 


250607 


41 


20 


690098 


375 


940409 


118 


749089 


493 


250311 


40 


21 


9.690323 


374 


9.940338 


118 


9.749985 


493 


10.250015 


39 


22 


690548 


374 


940237 


118 


750281 


492 


249719 


38- 


23 


690772 


374 


940195 


118 


750576 


492 


249424 


37 


24 


690996 


374 


940125 


119 


750872 


492 


249128 


35 


25 


691220 


373 


940054 


119 


751167 


492 


248333 


35 


26 


691444 


373 


939982 


119 


751452 


492 


248.538- 


34 


27 


691668 


373 


939911 


119 


751757 


492 


248243 


33 


28 


691892 


373 


939340 


119 


752052 


491 


247948 


32 


29 


692115 


372 


939768 


119 


752347 


491 


247653 


31 


30 


692339 


372 


939697 


119 


752342 


491 


247358 


30 


31 


9.692562 


372 


9.939525 


119 


9.7.52937 


491 


10.247053 


29 


32 


692785 


371 


939554 


119 


753231 


491 


245759 


28 


33 


693003 


371 


939482 


119 


753520 


491 


246474 


27 


34 


693231 


371 


939410 


119 


753820 


490 


246180 


26 


35 


693453 


371 


939339 


119 


754115 


490 


245885 


25 


36 


693676 


370 


939267 


120 


754409 


490 


245591 


24 


37 


6938y8 


370 


939195 


120 


754703 


490 


245297 


23 


38 


694120 


370 


939123 


120 


754997 


400 


245003 


22 


39 


694342 


370 


930052 


120 


755291 


490 


244709 


21 


40 


694564 


369 


938980 


120 


755585 


489 


244415 


20 


41 


9.694788 


389 


9.938908 


120 


9.755878 


489 


10.244122 


19 


42 


695007 


369 


938836 


120 


756172 


489 


243828 


18 


43 


695229 


369 


938763 


120 


756405 


489 


243535 


17 


44 


695450 


368 


938691 


120 


755759 


489 


243241 


16 


45 


695671 


358 


938319 


120 


757052 


439 


242948 


15 


46 


695892 


368 


938547 


120 


757345 


488 


242655 


14 


47 


69G113 


368 


638475 


120 


757638 


488 


242352 


13 


48 


695334 


367 


938402 


121 


757931 


488 


242059 


12 


49 


696554 


367 


938330 


121 


758224 


488 


241776 


11 


50 


693775 


367 


938258 


121 


758517 


488 


241483 


10 


51 


9.696995 


367 


9.938185 


121 


9.758810 


488 


10.241190 


9 


52 


697215 


366 


938133 


121 


759102 


487 


240898 


8 


53 


697435 


366 


938040 


121 


759395 


487 


240605 


7 


54 


697654 


366 


937957 


121 


759687 


487 


240313 


6 


55 


697874 


366 


937895 


121 


759979 


487 


240021 


5 


56 


693094 


365 


937822 


121 


760272 


487 


239728 


4 


57 


693313 


365 


937749 


121 


760554 


487 


239436 


3 


58 


693532 


365 


937676 


121 


760856 


486 


239144 


2 


59 


693751 


365 


937604 


121 


761148 


486 


238852 


1 


60 


698970 


364 


937531 


121 


761439 


486 


238561 







Cosine 




Sine 




Cotang. 




Tang. 


M 



60 Degrees. 



74 




(30 Degrees.) a 


TABLE OF LOGARITHMIC 






M. 


i Sine 


1 D 


1 Cosine 


! D. 


1 Tang. 


1 D. 


1 Cotang. 


1 







9.61J8970 


364 


9.937531 


121 


9.761439 


486 


10.238561 


60 




1 


699189 


364 


937458 


122 


761731 


486 


238269 


59 




2 


699407 


364 


937385 


122 


762023 


486 


237977 


58 




3 


699626 


364 


937312 


122 


762314 


486 


237686 


57 




4 


699844 


363 


937238 


122 


762606 


485 


237394 


56 




5 


700062 


363 


937165 


122 


762897 


485 


237103 


55 




6 


700280 


363 


937092 


122 


763188 


485 


236812 


54 




7 


700498 


363 


937019 


122 


763479 


485 


23G521 


53 




8 


700716 


363 


93(5946 


122 


763770 


485 


236230 


52 




9 


700933 


362 


936872 


122 


764061 


485 


235939 


51 




10 


701151 


362 


936799 


122 


764352 


484 


235648 


50 




11 


9.701368 


362 


9.936725 


122 


9.764643 


484 


10.235357 


49 




12 


701585 


362 


936052 


123 


764933 


484 


2350G7 


48 




13 


701802 


361 


936578 


123 


765224 


484 ' 


234776 


47 




14 


702019 


361 


936505 


123 


765514 


484 


234486 


46 




15 


702236 


361 


936431 


123 


765805 


484 


234195 


45 




16 


702452 


361 


936357 


123 


766095 


484 


233905 


44 




17 


702669 


360 


936284 


123 


766385 


483 


233615 


43 




18 


702885 


300 


936210 


123 


766075 


483 


233325 


42 




19 


703101 


360 


936136 


123 


76r965 


483 


233035 


41 




20 


703317 


360 


936002 


123 


767255 


483 


232745 


40 




21 


9.703533 


359 


9.935988 


123 


9.767545 


483 


10.232455 


39 




22 


703749 


359 


935914 


123 


767834 


483 


232166 


38 




23 


703964 


359 


935840 


123 


7C8124 


482 


231876 


37 




24 


704179 


359 


935766 


124 


7G8413 


482 


231587 


36 




25 


704395 


359 


935G92 


124 


7G8703 


482 


231297 


35 




26 


704010 


358 


935618 


124 


768992 


482 


231008 


34 




27 


704825 


358 


935543 


124 


769281 


482 


230719 


33 




28 


705040 


358 


9354G9 


124 


769570 


482 


230430 


32 




29 


705254 


358 


935395 


124 


769860 


481 


230140 


31 




30 


705469 


357 


9353-20 


124 


770148 


481 


229852 


30 




31 


9.705683 


357 


9.935246 


124 


9.770437 


481 


10.229563 


29 




32 


705898 


357 


935171 


124 


770726 


481 


229274 


28 




33 


700112 


357 


935097 


124 


771015 


481 


228985 


27 




34 


706326 


356 


935022 


124 


771303 


481 


228697 


26 




35 


706539 


356 


934948 


124 


771592 


481 


228408 


25 




36 


706753 


356 


934873 


124 


771880 


480 


228120 


24 




37 


706967 


356 


934798 


125 


772168 


480 


227832 


23 




38 


707180 


355 


934723 


125 


772457 


48;) 


227543 


22 




39 


707393 


355 


934649 


125 


772745 


480 


227255 


21 




40 


707606 


355 


934574 


125 


773033 


480 


226967 


20 




41 


9.707819 


355 


9.934499 


125 


9.773321 


480 


10.226679 


19 




42 


708032 


354 


934424 


125 


773()C8 


479 


226392 


18 




43 


708245 


354 


934349 


125 


773896 


479 


226104 


17 




44 


708458 


354 


934274 


125 


774184 


479 


225816 


16 




45 


708070 


354 


934199 


125 


774471 


479 


225529 


15 




46 


708882 


353 


934123 


125 


774759 


479 


225241 


14 




47 


709094 


353 


934048 


125 


775046 


479 


224954 


13 




48 


709306 


353 


933973 


125 


775333 


479 


224667 


12 




49 


709518 


353 


933898 


120 


775621 


478 


224379 


11 




50 


709730 


353 


933822 


126 


775908 


178 


^24092 


10 




51 


9.709941 


352 


9.933747 


126 


9.776195 


4:8 


10.223805 


9 




52 


710153 


352 


'^9331)71 


126 


776482 


478 


223518 


8 




53 


710364 


352 


933595 


126 


776769 


478 


223231 


7 




54 


710575 


352 


933520 


126 


777055 


478 


222945 


6 




55 


710786 


351 


933445 


126 


777342 


478 


222C58 


5 




56 


710997 


351 


9333<]9 


126 


777()28 


477 


222372 


4 




57 


7112C8 


351 


933293 


126 


777915 


477 


222085 


3 




58 


711419 


351 


933217 


126 


778201 


477 


221799 


2 




59 


7 J 1629 


350 


933141 


126 


778487 


477 


221512 


1 




CO 


711839 


350 


9330(i() 


126 


778774 


477 


221226 







,, 


Cosine 




Sine 1 




Cotang. 




Tang. 


M. 1 





59 Degrees. 







SINES 


AND TANGENTS 


. (31 Degrees.) 




75 


► 


M 


Sine 


!>• 


Cosine 


D. 


Tang. 


D- 


( Cotang. 









9.711839 


350 


9.9330G6 


126 


9.778774 


477 


10.221226 


60 




1 


712050 


350 


932990 


127 


779060 


477 


220940 


59 




O 


7122G3 


350 


932914 


127 


779346 


476 


220054 


58 




3 


712409 


349 


93-2833 


127 


779632 


476 


220368 


57 




4 


712679 


349 


932762 


127 


779918 


476 


220082 


56 




5 


712889 


349 


932385 


127 


780203 


476 


219797 


55 




6 


713098 


349 


932309 


127 


780483 


476 


219511 


54 




7 


713308 


349 


93253:i 


127 


780775 


476 


219225 


53 




8 


713517 


348 


932457 


127 


781030 


476 


218940 


52 




9 


713723 


348 


932380 


127 


781346 


475 


218654 


51 




10 


713935 


348 


932304 


127 


781631 


475 


218369 


50 




11 


9.714144 


348 


9.932228 


127 


9.781915 


475 


10.218384 


49 




12 


714352 


347 


932151 


127 


782202 


475 


217739 


48 




13 


714561 


347 


932075 


128 


732486 


475 


217514 


47 




14 


714769 


347 


931998 


128 


782771 


475 


217229 


46 




15 


714978 


347 


931921 


128 


783056 


475 


213944 


45 




16 


715183 


347 


931845 


123 


783341 


475 


216359 


44 




17 


715394 


346 


931768 


128 


783G2B 


474 


216374 


43 




18 


715302 


343 


931391 


128 


783910 


474 


216090 


42 




19 


715809 


346 


931614 


128 


784195 


474 


215805 


41 




23 


716017 


346 


931537 


128 


781479 


474 


215521 


40 




21 


9.716224 


345 


9.931460 


128 


9.784764 


474 


10.215236 


39 




22 


716432 


345 


931383 


128 


785048 


474 


2149.52 


38 




23 


716639 


345 


931306 


123 


785332 


473 


214338 


37 




24 


716846 


345 


931229 


129 


735616 


473 


214384 


36 




25 


717053 


345 


931152 


129 


785900 


473 


214100 


35 




2G 


717259 


344 


931075 


129 


786184 


473 


213816 , 


34 




27 


717466 


344 


930998 


129 


786468 


473 


213532 


33 




23 


717673 


344 


930921 


129 


786752 


473 


213248 


32 




29 


717879 


344 


930843 


129 


787033 


473 


212964 


31 




30 


718085 


343 


930766 


129 


787319 


472 


212681 


30 




31 


9.71S291 


343 


9.933688 


129 


9.787603 


472 


10.212397 


29 




32 


718497 


343 


930611 


129 


787886 


472 


212114 


28 




33 


718703 


343 


930533 


129 


788170 


472 


211830 


27 




34 


718909 


343 


930458 


129 


788453 


472 


211547 


26 




35 


719114 


342 


933378 


129 


788738 


472 


211264 


25 




36 


719320 


342 


930303 


130 


789019 


472 


210981 


24 




37 


719525 


342 


930223 


130 


789302 


471 


210698 


23 




38 


719730 


342 


933145 


130 


789585 


471 


210415 


22 




39 


719935 


341 


930367 


130 


789868 


471 


210132 


21 




40 


720140 


341 


929989 


130 


790151 


471 


209349 


20 




41 


9.720345 


341 


9.929911 


130 


9.790433 


471 


10.209567 


19 




42 


720549 


341 


929833 


130 


790716 


471 


209234 


18 




43 


720754 


340 


929755 


130 


790999 


471 


200001 


17 




44 


720958 


340 


929677 


130 


791281 


471 


208719 


16 




45 


721162 


340 


929599 


]30 


791533 


470 


208437 


15 




46 


721366 


340 


929521 


130 


791843 


470 


208154 


14 




47 


721570 


340 


929442 


130 


792128 


470 


207872 


13 




48 


721774 


339 


929364 


131 


792410 


470 


207590 


12 




49 


721978 


339 


929283 


131 


792692 


470 


207303 


11 




50 


722181 


339 


929207 


131 


792974 


470 


207026 


10 




51 


9.722385 


339 


9.929129 


131 


9.793256 


470 


10.206744 


9 




52 


722588 


339 


929050 


131 


793533 


4()9 


206462 


8 




53 


722791 


338 


928972 


131 


703819 


439 


206181 


7 




54 


722994 


338 


928893 


131 


794101 


469 


205899 


6 




55 


723197 


338 


928815 


131 


794333 


469 


205617 


5 




56 


723400 


338 


928733 


131 


794664 


469 


205336 


4 




57 


723303 


337 


928357 


131 


794915 


439 


205055 


3 




58 


723805 


337 


928578 


131 


795227 


469 


204773 


2 




59 


724007 


337 


928499 


131 


795508 


468 


204492 


1 




60 


724210 


337 


928420 


131 


795789 


46« 


204211 









(Cosine 


i 


1 Sine 


1 


1 Cotang. 


1 


1 Tang. 


1 M. 





58 Degrees. 



76 


(32 Degrees.) a 


TABLE OF LOGARITHMIC 






M. 


Sine 


D 


Cosine 


1 D. 


1 Tang. 


1 J). 


1 Cotang. 


1 







9. 7242 JO 


337 


9.928420 


132 


9.79.5789 


468 


10.204211 


60 




1 


724412 


337 


928342 


132 


796070 


468 


203930 


59 




2 


724614 


336 


928263 


132 


796351 


468 


203649 


58 




3 


724816 


336 


928183 


132 


796632 


468 


203368 


57 




4 


725017 


336 


928104 


132 


796913 


468 


• 203087 


56 




5 


725219 


336 


928025 


132 


797194 


4(58 


202806 


55 




C 


725420 


335 


927946 


132 


797475 


468 


202525 


54 




7 


725(522 


335 


927867 


i:}2 


797755 


468 


202245 


53 




8 


725823 


335 


927787 


132 


798036 


467 


201964 


I 52 




9 


726024 


335 


927708 


132 


798316 


467 


201684 


51 




10 


726225 


335 


927629 


132 


798596 


467 


201404 


50 




11 


9.726426 


334 


9.927549 


132 


9.798877 


467 


10.201123 


49 




12 


726626 


334 


927470 


133 


799157 


467 


200843 


48 




13 


726827 


334 


927390 


133 


799437 


467 


200563 


47 




14 


727027 


334 


927310 


133 


799717 


467 


200283 


46 




15 


727228 


334 


927231 


133 


799997 


466 


200003 


45 




16 


727428 


333 


927151 


133 


800277 


466 


199723 


44 




17 


727028 


333 


927071 


133 


800557 


466 


199443 


43 




18 


727828 


333 


926991 


133 


800836 


466 


199164 


42 




19 


728027 


333 


926911 


133 


801116 


466 


198884 


41 




20 


728227 


333 


926831 


133 


801390 


466 


198604 


40 




21 


9.728427 


332 


9.926751 


133 


9.801675 


466 


10.198325 


39 




1 o2 


728626 


332 


926671 


133 


801955 


460 


198045 


38 




23 


728825 


332 


926591 


133 


802234 


465 


197766 


37 




24 


729024 


332 


926511 


134 


802513 


465 


197487 


36 




25 


729223 


331 


926431 


134 


802792 


465 


197208 


35 




26 


729422 


331 


926351 


134 


803072 


465 


196928 


34 




27 


729(>21 


331 


926270 


134 


803351 


465 


196649 


33 




28 


729820 


331 


926190 


134 


803630 


465 


196370 


32 




29 


7300 J 8 


330 


926110 


134 


803908 


465 


196092 


31 




30 


730216 


330 


926029 


134 


804187 


465 


195813 


30 




31 


9.730415 


330 


9.925949 


134 


9.804466 


464 


10.195534 


29 




32 


730613 


330 


925868 


134 


804745 


464 


195255 


28 




33 


73081 1 


330 


925788 


134 


805023 


464 


194977 


27 




34 


731009 


329 


925707 


134 


805302 


4(54 


194698 


20 




35 


731206 


329 


925626 


134 


805580 


464 


194420 


25 




36 


731404 


329 


925545 


135 


805«59 


464 


194141 


24 




37 


731602 


329 


925465 


135 


806137 


464 


193863 


23 




38 


731799 


329 


925:^84 


135 


806415 


4(53 


193585 


22 




39 


731996 


328 


925303 


135 


806693 


463 


193307 


oi 




40 


732193 


328 


925222 


135 


806971 


403 


I93li29 


20 




41 


9.732390 


328 


9.925141 


135 


9.807249 


463 


10.192751 


19 




42 


732587 


328 


925060 


135 


807527 


4(53 


192473 


18 




43 


732784 


328 


924979 


135 


807805 


4(53 


192195 


17 




44 


732980 


327 


924897 


135 


808083 


463 


191917 


16 




45 


733177 


327 


924816 


135 


808361 


463 


191639 


15 




46 


733373 


327 


924735 


136 


808(538 


462 


191362 


14 




47 


733569 


327 


924654 


136 


808916 


462 


191084 


13 




48 


733765 


327 


924572 


136 


809193 


4(52 


190807 


12 




49 


733961 


326 


924491 


136 


809471 


462 


190529 


11 




50 


734157 


326 


924409 


130 


809748 


462 


190252 


10 




51 


9.734353 


326 


9.924328 


136 


9.810025 


462 


10.189975 


9 




52 


734549 


32r, 


924246 


136 


810302 


462 


189698 


8 




53 


7.34744 


325 


924164 


136 


810580 


462 


189420 


7 




54 


734939 


325 


924083 


136 


810857 


4(52 


189143 


6 




55 


735135 


325 


92400 1 


136 


811134 


461 


18886(5 


5 




56 


735330 


325 


923919 


136 


811410 


461 


188590 


4 




57 


735525 


325 


923837 


136 


811687 


461 


188313 


3 




58 


735719 


324 


923755 


137 


8119(54 


461 


188030 


2 




5i) 


735914 


324 


923(573 


137 


812241 


461 


187759 


1 




(iO 


736109 


324 


923591 


137 


812517 


461 


187483 









1 Cosine 


1 


1 Sine 





Cotang. 




Tang. 


M. 





57 Degrees. 







SINES 


AND TANGENTS 


. (33 Degrees.) 




77 


1 M. 1 


Sine 1 


D. ( 


Cosine 


D. ( 


Tang. 1 


D- 1 


Cotang 1 







9.736109 


324 


9.9-23591 


137 


9.812517 


461 


10.187482 


60 


1 


736303 


324 


923509 


137 


812794 


461 


187206 


59 


2 


736498 


324 


923427 


1.37 


813070 


461 


186930 1 


58 ; 


3 


736692 


323 


923345 


137 


813347 


460 


186653 


57 


4 


736386 


323 


923263 


137 


813623 


460 


186377 


56 


5 


737080 


323 


923181 


137 


813899 


460 


186101 


55 


6 


737274 


323 


923098 


137 


814175 


460 


185825 


54 i 


7 


737467 


323 


923016 


137 


814452 


460 


185548 


53 = 


8 


737661 


322 


922933 


137 


814728 


460 


185272 


52 


9 


737855 


322 


922851 


137 


815004 


460 


184996 


51 ■ 


10 


738048 


322 


9227G8 


138 


81.5279 


460 


184721 


50 


11 


9.738241 


322 


9.92-2686 


238 


9.81.5555 


459 


10.184445 


49 ■ 


12 


738434 


322 


922603 


138 


815831 


459 


184109 


48 ; 


13 


738627 


321 


922520 


138 


816107 


459 


183893 


47 


14 


738820 


321 


922438 


138 


816382 


459 


183618 


46 


15 


739013 


321 


922355 


138 


816658 


459 


183342 


45 


16 


739206 


321 


922272 


1.38 


816933 


459 


183067 


44 


17 


739398 


321 


922189 


138 


817209 


459 


182791 


43 


18 


739590 


320 


922106 


138 


817484 


459 


182516 


42 


19 


739783 


320 


922023 


138 


817759 


459 


182241 


41 


20 


739975 


320 


921940 


138 


818035 


458 


181965 


40 


21 


9.740167 


320 


9.921857 


139 


9.818310 


458 


10.181690 


39 1 


22 


740359 


3'K) 


021774 


139 


818585 


458 


181415 


38 


23 


740550 


319 


921691 


139 


8188G0 


458 


181140 


37 


24 


740742 


319 


921607 


139 


819135 


458 


180865 


36 


25 


740934 


319 


921524 


139 


819410 


458 


180590 


35 


26 


741125 


319 


921441 


139 


819G84 


458 


180316 


34 


27 


741316 


319 


921357 


139 


819959 


458 


180041 


33 


28 


741508 


318 


921274 


139 


820234 


458 


179766 


32 


29 


741699 


318 


921190 


139 


820508 


457 


179492 


31 


30 


741889 


318 


921107 


139 


820783 


457 


179217 


30 


31 


9.742080 


318 


9.921023 


139 


9.821057 


457 


10.178943 


29 


32 


742271 


31« 


920939 


140 


821332 


457 


178668 


28 


33 


742462 


317 


920856 


140 


821606 


4.57 


178394 


27 


34 


742652 


317 


920772 


140 


821880 


457 


1781-20 


26 


35 


742842 


317 


920688 


140 


822154 


357 


177846 


25 


36 


743033 


317 


920604 


140 


822429 


457 


177571 


24 


37 


743223 


317 


920520 


140 


822703 


457 


177297 


23 


38 


743413 


31 6 


920436 


140 


822977 


456 


177023 


22 


39 


743602 


316 


9203.52 


140 


823250 


4.56 


176750 


21 


40 


743792 


316 


920268 


140 


823524 


456 


176476 


20 


41 


9.743982 


316 


9.920184 


140 


9.82.3798 


4.56 


10.176202 


19 


42 


744171 


316 


920099 


140 


824072 


456 


175928 


18 


43 


7443f;i 


315 


920015 


140 


824345 


456 


175655 


17 


1 44 


744550 


315 


919931 


141 


824619 


456 


175381 


16 


45 


744739 


315 


919846 


141 


824893 


456 


175107 


15 


46 


744928 


315 


919762 


141 


825166 


456 


174834 


14 


47 


745117 


315 


919677 


141 


82.5439 


455 


174561 


13 


48 


745306 


314 


919593 


141 


825713 


455 


174287 


12 


49 


745494 


314 


919508 


141 


825986 


455 


174014 


11 


50 


745G83 


314 


919424 


141 


826259 


455 


173741 


10 


51 


9.745871 


314 


9.919339 


141 


9.820532 


455 


10.173468 


9 


52 


746059 


314 


919-254 


141 


826805 


455 


173195 


8 


53 


746248 


313 


919169 


141 


827078 


4.55 


172922 


7 


54 


740436 


313 


91 9085 


141 


827351 


455 


172649 


6 


55 


746024 


313 


919000 


141 


827624 


455 


172376 


5 


56 


746812 


313 


918915 


142 


827897 


454 


172103 


4 


57 


74r999 


313 


918830 


142 


828170 


454 


171830 


3 


58 


747187 


312 


918745 


142 


828442 


454 


171558 


2 


1 59 


747374 


312 


918659 


142 


828715 


4.54 


171285 


1 


I 60 


7475G2 


312 


918.574 


142 


828987 


454 


171013 







1 Cosine 


1 


1 Sine 


1 


! Cotang. 


1 


1 Tang. 


1 M. 








56 


Degr< 


ees. 









78 




(34 Degrees.) a 


TABLE OF LOGARITHMIC 




M. 


1 Sine 


1 D. 


1 Cosine 


1 D. 


Tang. 


D. 


Cotang. 


1 1 





9.747562 


312 


9.918.574 


142 


9.828987 


454 


10.171013 


60 


1 


747749 


312 


918489 


142 


829260 


454 


170740 


59 


2 


7479:J6 


312 


918404 


142 


820532 


454 


170468 


58 


3 


748123 


311 


918318 


142 


829305 


454 


170195 


57 


4 


748310 


311 


9182.33 


142 


830077 


4.54 


1(;9923 


56 


5 


748497 


311 


918147 


142 


83i.349 


453 


169651 


55 


6 


74S633 


311 


918)62 


142 


830021 


4.53 


169379 


54 


7 


748870 


311 


917976 


143 


830393 


453 


169107 


53 


8 


74905;i 


310 


917891 


143 


831165 


4.53 


168835 


52 


9 


749243 


310 


917805 


143 


831437 


453 


168563 


51 


10 


749429 


310 


917719 


143 


83176J 


453 


168291 


50 


11 


9.749315 


310 


9.917634 


143 


9.831981 


4.53 


10.168019 


49 


12 


749dJl 


310 


917.548 


143 


8322.53 


4.5'i 


167747 


48 


13 


749^87 


309 


917462 


143 


832525 


453 


167475 


47 


14 


7.30172 


309 


917376 


143 


832796 


4.53 


167204 


46 


15 


750:558 


309 


917200 


143 


833008 


4.52 


166932 


45 


16 


750r)4J 


3i)9 


917204 


143 


833339 


452 


166661 


44 


17 


750729 


309 


917118 


144 


833611 


452 


166389 


43 


18 


750314 


308 


917032 


144 


833882 


452 


166118 


42 


19 


751099 


308 


916940 


144 


834154 


452 


16584() 


41 


20 


751284 


308 


916859 


144 


834425 


452 


165575 


40 


21 


9.751469 


308 


9.916773 


144 


9.834696 


452 


10.165304 


39 


22 


751054 


308 


91{)687 


144 


834967 


4.52 


16.5033 


38 


23 


751839 


308 


916600 


144 


835238 


452 


164762 


37 


24 


752023 


307 


916514 


144 


83.5509 


452 


164491 


36 


25 


752208 


307 


916427 


144 


835783 


451 


164-220 


35 


2(5 


752392 


3)7 


916341 


144 


836)51 


451 


163949 


34 


27 


752576 


307 


916254 


144 


836322 


451 


163678 


aj 


28 


752760 


307 


916167 


145 


836593 


451 


163407 


32 


29 


752944 


306 


916081 


145 


833864 


451 


163136 


31 


30 


753128 


306 


915994 


145 


837134 


451 


162866 


30 


31 


9.753312 


306 


9.915907 


145 


9.837405 


451 


10.162.595 


29 


32 


7534iJ5 


306 


9.5^20 


145 


837675 


45i 


162325 


28 


33 


753679 


303 


915733 


145 


837946 


451 


162054 


27 


34 


753862 


3J5 


915646 


145 


838216 


451 


161784 


26 


35 


754046 


305 


91.5559 


145 


838487 


450 


161513 


25 


36 


754229 


305 


915472 


145 


838757 


450 


16 J 243 


24 


37 


7544 J 2 


305 


915585 


145 


839027 


450 


160973 


23 


38 


754595 


305 


915297 


145 


839-297 


450 


160703 


22 


39 


754778 


304 


91.5210 


145 


839568 


450 


160432 


21 


40 


754960 


304 


915123 


146 


839838 


450 


160162 


20 


41 


9.755143 


304 


9.915035 


146 


9.840108 


450 


10.159892 


19 


42 


755326 


304 


914948 


146 


840378 


450 


1.59622 


18 


43 


755503 


304 


914860 


146 


840647 


4.50 


159353 


17 


44 


755600 


304 


914773 


146 


840917 


449 


159083 


16 


45 


755372 


303 


914685 


146 


841187 


449 


1. 588 13 


15 


46 


756054 


303 


914598 


146 


8414.57 


449 


1.58543 


14 


47 


756236 


303 


914510 


146 


841726 


449 


1,58274 


13 


48 


756418 


303 


914422 


146 


841996 


449 


158004 


12 


49 


756600 


303 


914334 


146 


842266 


449 


157734 


11 


50 


756782 


302 


914246 


147 


842535 


449 


157465 


10 


51 


9.756963 


302 


9.914158 


147 


9.84-280.5 


449 


10.1.57195 


9 


52 


7.57144 


302 


914070 


147 


843074 


449 


156926 


8 


53 


757326 


302 


913982 


147 


8 13343 


449 


156657 


7 


54 


7575)7 


302 


913894 


147 


843612 


449 


156388 


6 


55 


757688 


301 


913800 


147 


84.3882 


448 


156118 


5 


56 


757869 


301 


913718 


147 


844151 


448 


15.5849 


4 


57 


758050 


301 


913630 


147 


844420 


448 


155.580 


3 


58 


758230 


301 


913.541 


147 


844689 


448 


15.5311 


i> 


59 


7,584 1 1 


301 


913453 


147 


844958 


448 


155042 


1 


60 


758591 


301 


913365 


147 


84.5227 


448 


1.54773 




_M., 




Cosine 




Sine 




Cotang. 




Tang. 



55 Degrees. 







SINE-S 


AND TANGENTS 


. (35 Degrees.) 




79 


M 


Sine 


D- 


Cosine 


D. 


1 Tang. 


1 D- 


1 Cotang. 


1 





9.758591 


301 


9.9i:33G5 


147 


9.815227 


448 


10 154773 


60 




753772 


300 


913273 


147 


845496 


448 


154504 


59 


2 


753952 


300 


913187 


148 


815764 


448 


154236 


58 


3 


759132 


300 


913099 


143 


846033 


448 


153937 


57 


4 


759312 


300 


913010 


148 


843302 


448 


153698 


56 


5 


759492 


300 


912922 


148 


84(5570 


447 


153430 


55 


G 


759672 


299 


912333 


143 


846S39 


447 


153161 


54 


7 


759852 


299 


912744 


148 


847107 


447 


15Sv593 


53 


8 


760331 


299 


912655 


143 


847376 


447 


152G24 


52 


9 


760211 


299 


912536 


118 


84764 1 


447 


152356 


51 


10 


760390 


289 


912477 


148 


847913 


447 


152087 


50 , 


11 


9.760569 


298 


9.912383 


148 


9.848181 


447 


10.151819 


49 


12 


760748 


298 


912299 


149 


818449 


447 


151551 


48 


13 


7G0927 


293 


912210 


149 


848717 


447 


151283 


47 


14 


761106 


293 


912121 


149 


848983 


447 


151014 


46 


15 


751285 


298 


912031 


149 


849 J54 


447 


150746 


45 


16 


761464 


293 


911942 


149 


819522 


447 


150478 


44 


17 


761G42 


297 


911853 


149 


849793 


446 


150210 


43 


18 


761821 


2S7 


911763 


149 


853058 


446 


149942 


42 


19 


761999 


297 


911674 


149 


853325 


446 


149675 


41 


20 


762177 


297 


911584 


149 


853593 


446 


149407 


40 


21 


9.762353 


297 


9.911495 


149 


9.850361 


446 


10.149139 


39 


22 


762534 


295 


911405 


149 


851129 


446 


148371 


38 


23 


762712 


296 


911315 


150 


851396 


446 


148604 


37 


24 


762889 


293 


911226 


150 


851664 


446 


143336 


36 


25 


7630G7 


296 


911136 


150 


851931 


446 


148069 


35 


2J 


7C3245 


293 


911046 


150 


852193 


446 


147801 


34 


27 


7G3422 


293 


919956 


150 


852466 


446 


147534 


33 


23 


763G00 


295 


910866 


150 


8.>2733 


445 


147267 


32 


29 


7G3777 


295 


91')776 


150 


853301 


445 


146999 


31 


30 


763954 


295 


910683 


150 


853238 


445 


146732 


30 


31 


9.764131 


295 


9.910596 


150 


9.853535 


445 


10.143465 


29 


32 


764308 


295 


910503 


1.50 


853802 


445 


146198 


28 


33 


7G4485 


294 


910415 


150 


854069 


445 


145931 


27 


34 


764662 


294 


910325 


151 


854336 


445 


145664 


26 


35 


764838 


294 


910235 


151 


854303 


445 


145397 


25 


30 


7()5J15 


294 


910144 


151 


854870 


445 


145130 


24 


37 


765191 


294 


910054 


151 


855137 


445 


114863 


23 


38 


7653G7 


294 


909963 


151 


855404 


445 


144593 


22 


39 


765544 


293 


909873 


151 


855G71 


444 


144329 


21 


40 


765720 


293 


909782 


151 


855938 


444 


144032 


20 


41 


9.765896 


293 


9.909191 


151 


9.856204 


444 


10.143795 


19 


42 


766072 


293 


909 -.01 


151 


85G471 


444 


143529 


18 


43 


766247 


293 


909510 


151 


856737 


444 


143263 


17 


44 


766423 


293 


• 909419 


151 


857004 


444 


142998 


16 


45 


765598 


292 


9003-28 


152 


857270 


444 


142730 


15 


46 


766774 


292 


909237 


1.52 


857537 


444 


142463 


14 


47 


76:)949 


292 


909146 


1.52 


857803 


444 


142197 


13 


48 


767124 


292 


909055 


152 


858069 


444 


141931 


12 


49 


767300 


292 


903964 


152 


858336 


444 


141664 


11 


50 


767475 


' 291 


903873 


152 


853602 


443 


141398 


10 


51 


9.767649 


291 


9.903781 


152 


9.8.58868 


443 


10.141132 


9 


52 


767824 


291 


908690 


152 


859134 


443 


140866 


8 


53 


767999 


291 


908599 


152 


859400 


443 


143600 


7 


54 


768173 


291 


903.507 


152 


859866 


443 


140334 


6 


55 


768348 


290 


9)8416 


153 


859932 


443 


140068 


5 


56 


768522 


290 


908324 


153 


830193 


443 


139802 


4 


57 


768697 


290 


903233 


153 


860464 


443 


139536 


3 


58 


76887] 


290 


903141 


153 


860730 


443 


139270 


2 


59 


769045 


290 


938049 


153 


830995 


443 


139005 


1 


60 


769219 


290 


907958 


1.53 


861261 


443 


138739 







Ck)sine 




Sine 




Cotang. 




Tang. 


1 M. 



54 Degrees. 



80 



(36 Degrees.) a table op logarithmic 



ns: 


( Sine 


1 D 


1 Cosine 


1 D. 


( Tang. 


' D. 


1 CotoJjg. 


1 







9.709219 


290 


9.907958 


153 


9.861261 


443 


10.138739 


60 




1 


7G9393 


289 


907865 


1.53 


861527 


443 


138473 


59 




2 


7()95G6 


289 


907774 


153 


861792 


442 


138208 


58 




3 


769740 


289 


907682 


1.53 


862058 


442 


137942 


57 




4 


769913 


289 


907590 


1.53 


862323 


442 


137677 


56 




5 


770087 


289 


907498 


153 


862589 


442 


137411 


55 




6 


770260 


288 


907406 


1.53 


862854 


442 


1.37146 


54 




7 


770433 


288 


907314 


154 


863119 


442 


136881 


53 




8 


770606 


288 


907222 


154 


863385 


442 


136615 


52 




9 


770779 


288 


9D7129 


154 


863^50 


442 


136350 


51 




10 


770952 


288 


907037 


154 


863915 


442 


136085 


50 




11 


9.771125 


288 


9.906945 


154 


9.864180 


442 


10.135820 


49 




12 


771298 


287 


900.852 


1.54 


8(14445 


442 


135555 


48 




13 


771470 


287 


906760 


154 


864710 


442 


135290 


47 




14 


771643 


287 


906667 


154 


864975 


441 


135025 


40 




15 


771815 


287 


906575 


1,54 


865240 


441 


134760 


45 




IG 


771987 


287 


906482 


154 


865505 


441 


134495 


44 




17 


772159 


287 


906389 


1.55 


865770 


441 


134230 


43 




18 


772331 


286 


906296 


155 


866035 


441 


133965 


42 




19 


772503 


286 


906204 


155 


86G300 


441 


133700 


41 




20 


772675 


280 


906111 


155 


866564 


441 


133436 


40 




21 


9.772847 


286 


9.900018 


155 


9.860829 


441 


10.133171 


39 




22 


773018 


286 


905925 


155 


867094 


441 


132906 


38 




23 


773190 


286 


905832 


155 


807358 


441 


132642 


37 




24 


773361 


285 


905739 


155 


867623 


441 


132377 


36 




25 


773533 


285 


905645 


155 


867887 


441 


132113 


35 




26 


773704 


285 


905552 


155 


868152 


440 


131848 


34 




27 


773875 


285 


905459 


155 


868416 


440 


131584 


33 




28 


774046 


285 


905366 


156 


868680 


440 


131320 


32 




29 


7742] 7 


285 


905272 


156 


868945 


440 


131055 


31 




30 


774388 


284 


905179 


156 


809209 


440 


130791 


30 




31 


9.774558 


284 


9.905085 


156 


9.869473 


440 


10.130527 


29 




32 


774729 


284 


904992 


1.56 


869737 


440 


130263 


28 




33 


7748:)9 


284 


904898 


L53 


870001 


440 


129999 


27 




34 


775070 


284 


901804 


1.56 


870265 


440 


1297.35 


26 




35 


775240 


284 


90471 1 


158 


8705v9 


440 


129471 


25 




30 


775410 


283 


904617 


156 


870793 


440 


129207 


24 




37 


775580 


283 


904523 


156 


871057 


440 


128943 


23 




38 


775750 


283 


904429 


157 


871.321 


440 


128679 


22 




39 


775920 


283 


904335 


157 


871585 


440 


128415 


21 




40 


776090 


283 


904241 


157 


871849 


439 


128151 


20 




41 


9.776259 


283 


9.904147 


157 


9.872112 


439 


10.127888 


19 




42 


776429 


282 


9;)40.73 


157 


872376 


439 


127624 


18 




43 


776598 


282 


903959 


157 


872()40 


439 


1273()0 


17 




44 


7767(;8 


282 


9:j:]8:)4 


1.57 


872903 


439 


127097 


16 




45 


776937 


282 


903770 


157 


873167 


439 


126833 


15 




46 


777106 


282 


9t)3()76 


157 


87.3430 


439 


126570 


14 




47 


77/275 


281 


903581 


157 


873694 


439 


126306 


13 




48 


777444 


281 


903487 


L57 


873957 


439 


12(5043 


12 




49 


777613 


281 


903:]92 


158 


874220 


439 


125780 


11 




50 


777781 


281 


903298 


158 


874484 


439 


125516 


10 




51 


9.///950 


281 


9.903203 


158 


9.874747 


439 


10.12,52,53 


9 




52 


778119 


281 


9o:ni;8 


1.58 


875010 


439 


124990 


8 




53 


778287 


280 


903014 


1.58 


875273 


438 


124727 


7 




54 


778455 


280 


902919 


1.58 


875536 


438 


124464 


6 




55 


778624 


280 


902824 


1,58 


875800 


438 


124200 


5 




56 


778792 


280 


902729 


1.58 


876003 


438 


123937 


4 




57 


7789C)0 


280 


902(534 


1.58 


876321) 


438 


123(574 


3 




58 


779128 


280 


902539 


159 


876.589 


438 


123411 


2 




59 


779295 


279 


902444 


L59 


876851 


438 


323149 


1 




CO 


779463 


279 


902349 


159 


877114 


4.38 


122886 







1 


CosineH 


1 


Sine 1 




Cotang. 




Tang. 


"mT 





53 Degrees 







SINES 


AND TANGENTS 


. (37 Degrees.) 




81 


M. 


Sine 


D. 1 


Cosine 


D. 


Tang. 1 


D 


Cotang. 







9.779463 


279 


9.902349 


159 


9.877114 


438 


10. 1228 -:6 


60 


1 


779631 


279 


902253 


159 


877377 


438 


12-2623 


59 


2 


779798 


279 


902158 


159 


877640 


433 


12-2300 


58 


3 


779966 


279 


902OG3 


159 


877903 


433 


122097 


57 


4 


780133 


279 


901957 


159 


878105 


438 


1-21335 


58 


1 ^ 


780300 


278 


901872 


159 


878428 


438 


121572 


55 


'• 6 


780467 


278 


901776 


159 


878891 


433 


121309 


54 


7 


780634 


278 


901681 


159 


878953 


437 


121047 


53 


8 


780831 


278 


9015S5 


159 


879216 


437 


120784 


52 


9 


780J68 


278 


901490 


159 


879478 


437 


120522 


51 


10 


781134 


278 


901394 


160 


879741 


437 


120259 


50 


11 


9.781301 


277 


9.901293 


160 


9.833003 


437 


10.119997 


49 


12 


781408 


277 


9J1202 


169 


88321)5 


437 


119735 


48 


13 


781634 


277 


901103 


160 


8S0528 


437 


119472 


47 


14 


781800 


277 


901010 


160 


880790 


437 


119210 


46 


15 


781936 


277 


930914 


160 


881052 


437 


118948 


45 


16 


78-2132 


277 


900818 


160 


881314 


437 


118688 


44 


17 


782-298 


276 


900722 


180 


88] 576 


437 


118424 


43 


18 


782464 


270 


900626 


160 


881839 


437 


118161 


42 


19 


782830 


276 


900529 


160 


882101 


437 


117899 


41 


20 


782796 


276 


90J433 


]61 


882363 


436 


117637 


40 


21 


9.782961 


276 


9.900337 


161 


9.832625 


436 


10.117375 


39 


22 


783127 


276 


900240 


161 


88-2887 


436 


117113 


38 


23 


783292 


275 


900144 


161 


883148 


436 


1168.52 


37 


24 


783458 


275 


900047 


161 


883410 


436 


116590 


35 


25 


783623 


275 


899951 


181 


883672 


433 


1163-28 


35 


26 


783788 


275 


899854 


161 


883934 


436 


116066 


34 


27 


783953 


275 


899757 


161 


884196 


436 


115804 


33 


28 
29 


784118 


275 


899360 


161 


834457 


436 


115543 


32 


784-282 


274 


899564 


161 


884719 


436 


115-281 


31 


30 


784447 


274 


899467 


162 


884980 


436 


1150-20 


30 


31 


9.784612 


274 


9.899370 


162 


9.885242 


438 


10.114758 


29 


32 


784778 


274 


899273 


162 


885503 


436 


114497 


28 


33 


784941 


274 


899176 


162 


885765 


438 


1 J 4235 


27 


34 


785105 


274 


899078 


162 


883026 


436 


113974 


28 


35 


785269 


273 


898981 


162 


886-283 


436 


113712 


25 


36 


785433 


273 


898884 


162 


883549 


435 


113451 


24 


37 


785597 


273 


898787 


162 


836810 


435 


113190 


23 


38 


785761 


273 


898889 


162 


887072 


435 


11-29-28 


22 


39 


785925 


273 


893592 


162 


837333 


435 


112507 


21 


40 


786089 


273 


898494 


163 


887594 


435 


112408 


20 


41 


9.786252 


272 


9.898397 


163 


9.837855 


435 


10.112145 


19 


42 


786416 


272 


898299 


163 


883116 


435 


111884 


18 


43 


786579 


272 


893202 


163 


888377 


435 


111623 


17 


44 


786742 


272 


898104 


163 


833639 


435 


111361 


16 


45 


786906 


272 


898008 


163 


888900 


435 


111100 


15 


46 


787069 


272 


897908 


163 


839160 


435 


110340 


14 


47 


787232 


271 


897810 


163 


889421 


435 


110579 


13 


48 


787395 


271 


897712 


163 


839882 


435 


110318 


12 


49 


787557 


271 


897614 


163 


889943 


435 


110057 


11 


50 


787720 


271 


897516 


163 


890-204 


434 


109796 


10 


51 


9.787883 


271 


9.897418 


164 


9.890465 


434 


10.109535 


9 


52 


788045 


271 


897320 


164 


890725 


434 


109-275 


8 


53 


788208 


27! 


897222 


164 


890936 


434 


109014 


7 


54 


788370 


270 


897123 


164 


891-247 


431 


103753 


6 


55 


78S532 


270 


897025 


164 


891507 


434 


103493 


5 


56 


788694 


270 


890926 


164 


891708 


434 


103-232 


4 


57 


788856 


270 


896828 


164 


892023 


434 


107972 


3 


58 


789018 


270 


890729 


164 


89-2-289 


434 


107711 


2 


59 


789180 


270 


896631 


164 


892549 


431 


107451 


1 


60 


789342 


269 


896532 


164 


89-2810 


434 


107190 







1 Cosine 




Sine 


1 


1 Cotang. 




1 Tang. 


M. 



52 Degrees. 

ellwood's test prob. — 6. 



82 



(38 Degrees.) a table of logarithmic 



M. 


1 Sine 


1 D. 


1 Cosine 


i D. 


1 Tang. 


! D. 


1 Cotang. 


1 


1 





9.789342 


1 269 


9.896532 


164 


9.892810 


434 


10.107190 


60 




1 


789504 


269 


896433 


165 


893070 


434 


106930 


59 







789665 


269 


896335 


165 


893331 


434 


106669 


58 




3 


789827 


269 


896236 


165 


89.3591 


434 


106409 


57 




4 


789988 


269 


896137 


165 


893851 


4?A 


106149 


56 




5 


790149 


269 


896038 


165 


894111 


434 


105889 


55 




6 


790310 


268 


895939 


165 


894371 


434 


10.5629 


54 




7 


790471 


268 


89.5840 


165 


891632 


433 


105368 


53 




8 


790632 


268 


895741 


165 


894892 


4.33 


105108 


52 




9 


790793 


268 


89.5641 


165 


895152 


4.33 


104848 


51 




10 


790954 


268 


895542 


165 


89.54i2 


433 


1045&S 


50 




11 


9.791115 


268 


9.895443 


166 


9.895672 


433 


10.104328 


49 




12 


791275 


267 


895343 


166 


895932 


433 


104068 


48 




13 


791436 


267 


895244 


166 


896192 


433 


103808 


47 




14 


791596 


267 


895145 


166 


896452 


433 


103.548 


46 




15 


791757 


267 


895045 


166 


896712 


4.33 


193288 


45 




16 


791917 


267 


894945 


166 


896971 


433 


103029 


44 




17 


792077 


267 


894846 


160 


897231 


433 


102769 


43 




18 


792237 


266 


894746 


166 


897491 


433 


102509 


42 




19 


792397 


266 


894646 


166 


897751 


433 


102249 


41 




20 


792557 


266 


894546 


166 


898010 


433 


101990 


40 




21 


9.792716 


260 


9.894446 


167 


9.898270 


433 


10.101730 


39 




22 


792876 


266 


894346 


107 


896530 


433 


101470 


38 




23 


793035 


266 


894246 


167 


8J8789 


433 


101211 


37 




24 


793195 


265 


894146 


167 


899049 


432 


100951 


36 




25 


793354 


2/35 


894046 


167 


899308 


4.32 


100692 


35 




25 


793514 


265 


893946 


167 


899568 


432 


100432 


34 




27 


793673 


265 


893846 


167 


899827 


432 


100173 


33 




28 


793832 


265 


893745 


167 


900086 


432 


099914 


32 




29 


793991 


265 


893645 


167 


900346 


4.32 


099054 


31 




30 


794150 


264 


893544 


167 


900605 


432 


099395 


30 




31 


9.794308 


264 


9.893444 


168 


9.900864 


432 


10.099136 


29 




32 


794467 


264 


893343 


1-8 


901124 


432 


098876 


28 




33 


794626 


2(54 


893243 


ins 


901,383 


432 


098617 


27 




34 


794784 


264 


893142 


168 


901642 


432 


098358 


26 




35 


794942 


264 


893041 


168 


901901 


432 


098099 


25 




3(5 


795101 


264 


892940 


168 


902160 


432 


097840 


24 




37 


795259 


263 


892839 


168 


902419 


4.32 


097581 


23 




38 


795417 


2(53 


892739 


168 


902679 


432 


097321 


22 




39 


795575 


2(53 


892638 


168 


902938 


432 


097062 


21 




40 


795733 


2()3 


892536 


168 


903197 


431 


096803 


20 




41 


9.795891 


263 


9.892435 


169 


9.903455 


431 


10.096545 


19 




42 


796049 


263 


892334 


1(59 


903714 


431 


096286 


18 




43 


796206 


263 


892233 


169 


903973 


431 


096027 


17 




44 


796364 


2G2 


892132 


169 


904232 


431 


095768 


16 




45 


796521 


262 


892030 


169 


904491 


431 


095509 


15 




46 


796679 


262 


891929 


169 


904750 


431 


095250 


14 




47 


796836 


262 


891827 


169 


905008 


431 


094992 


13 




48 


796993 


262 


891726 


169 


905267 


431 


094733 


12 




49 


7'.)7150 


261 


891624 


169 


905526 


431 


094474 


11 




50 


7973U7 


261 


891523 


170 


905784 


431 


094210 


10 




51 


9.797464 


261 


9.891421 


170 


9.906043 


431 


10.093957 


9 




52 


797621 


261 


891319 


170 


906302 


431 


093(598 


8 




53 


797777 


261 


891217 


170 


906560 


431 


093440 


7 




54 


797934 


2(51 


891 1 15 


170 


906819 


431 


093181 


6 




55 


798091 


261 


891013 


170 


907077 


431 


092923 


5 




56 


798247 


261 


890911 


170 


907336 


431 


092664 


4 




57 


798403 


260 


890809 


170 


9075!)4 


431 


092406 


3 




58 


798560 


2(50 


890707 


170 


907852 


431 


092148 


2 




59 


798716 


2(50 


890605 


170 


908111 


4.30 


091889 


1 




60 


798872 


260 


890503 


170 


908369 


430 


091631 


" i 




1 


Cosine | 


1 


Pine 1 


1 


Cotang. 1 


1 


Tang. 1 


M. 1 





51 Degrees 







SINRS 


AND TANGENTS 


. (39 Degrees.) 




83 


Tm'.' 


j Sine 


1 D. 


Cosine 


D- 


Tang. 


D 


Cotang. 


~ 





9.798872 


260 


9.890503 


170 


9.908369 


430 


10.091631 


60 


1 


799028 


260 


890400 


171 


908!;28 


430 


091372 


59 


o 


799 J 84 


2()0 


890298 


171 


908886 


430 


091114 


58 


3 


799339 


259 


890195 


171 


909144 


430 


090856 


57 


4 


799495 


2.59 


890093 


171 


909402 


430 


090598 


56 


5 


799H51 


259 


889990 


171 


909660 


4.30 


090340 


55 


6 


799806 


259 


889888 


171 


909918 


430 


090082 


54 


7 


799962 


2.59 


889785 


171 


910177 


430 


089823 


53 


8 


800117 


259 


889^182 


171 


910435 


430 


089565 


52 


9 


8002V2 


258 


889579 


171 


910693 


430 


089307 


51 


10 


800427 


258 


889477 


171 


910951 


430 


089049 


50 


11 


9.800582 


258 


9.889374 


172 


9.911209 


430 


10.088791 


49 


12 


800737 


258 


889271 


172 


911467 


430 


088533 


48 


13 


800892 


258 


889168 


172 


911724 


430 


088276 


47 


14 


801047 


258 


889064 


172 


911982 


430 


088018 


46 


15 


801201 


258 


888961 


172 


912240 


4.30 


087760 


45 


16 


801356 


257 


888858 


172 


912498 


430 


087502 


44 


17 


801.511 


257 


888755 


172 


912756 


430 


187244 


43 


18 


80W05 


257 


888651 


172 


913014 


429 


08G98() 


42 


19 


801819 


257 


888.548 


172 


913271 


429 


C86:29 


41 


20 


801973 


257 


888444 


173 


913529 


429 


086471 


40 


21 


9.802128 


257 


9.888341 


173 


9.913787 


429 


10.086213 


39 


22 


802282 


256 


888237 


173 


914044 


429 


085956 


38 


23 


802436 


256 


888134 


173 


914302 


429 


C85G98 


37 


24 


802589 


256 


888030 


173 


9145G0 


429 


085440 


36 


25 


802743 


256 


887926 


173 


914817 


429 


085183 


35 


26 


802897 


256 


887822 


173 


915075 


429 


084925 ■ 


34 


27 


803050 


256 


887718 


173 


915332 


429 


084668 


33 


28 


803204 


256 


887G14 


173 


915590 


429 


084410 


32 


29 


803357 


255 


887510 


173 


915847 


429 


0841.53 


31 


30 


803511 


255 


887406 


174 


916104 


429 


083896 


30 


31 


9.8030n4 


255 


9.887302 


174 


9.916362 


429 


10.083638 


29 


32 


803817 


255 


887198 


174 


916619 


429 


083G81 


28 


33 


803970 


255 


887093 


174 


916877 


429 


083123 


27 


34 


8 '4123 


2.55 


880989 


174 


917134 


429 


082866 


26 


35 


804276 


254 


886885 


174 


917391 


429 


" 082609 


25 


36 


804428 


254 


88G780 


174 


917648 


429 


082352 


24 


37 


804581 


254 


886676 


174 


917905 


429 


082095 


23 


38 


804734 


254 


886571 


174 


918163 


428 


081837 


22 


39 


804886 


254 


886466 


174 


918420 


428 


081580 


21 


40 


805039 


254 


886362 


175 


918677 


428 


081323 


20 


41 


9.805191 


254 


9.886257 


175 


9.918934 


428 


10.081066 


19 


42 


805343 


253 


886152 


175 


919191 


428 


080809 


18 


43 


805495 


253 


886047 


175 


919448 


428 


080552 


17 


44 


805647 


253 


885942 


175 


919705 


428 


080295 


16 


45 


805799 


253 


88.5837 


175 


919962 


428 


080038 


15 


46 


805951 


253 


885732 


175 


920219 


428 


079781 


14 , 


47 


806103 


253 


885627 


175 


920476 


428 


079524 


13 


48 


800254 


253 


885522 


175 


920733 


428 


079267 


12 


49 


806406 


252 


885416 


175 


920990 


428 


079010 


11 


50 


806557 


252 


885311 


176 


921247 


428 


078753 


10 


51 


9.806709 


252 


9.885205 


176 


9.921.503 


428 


10.078497 


9 


52 


806860 


252 


885100 


176 


921760 


428 


078240 


8 


53 


807011 


252 


884994 


176 


922017 


428 


077983 


7 


54 


807163 


252 


884889 


176 


922274 


428 


077726 


6 


55 


807314 


252 


884783 


176 


922530 


428 


077470 


5 


56 


807465 


251 


884677 


176 


922787 


428 


077213 


4 


57 


807615 


251 


884572 


176 


923044 


428 


076956 


3 


58 


807766 


251 


884466 


176 


923300 


428 


076700 


2 


59 


807917 


251 


884360 


176 


923557 


427 


076443 


1 


60 


808007 


251 


884254 


177 


923813 


427 


076187 


1 
M. i 




Cosine 




Sine 




Cotang. 




Tang 



50 Degrees. 



84 




^40 Degrees.) a 


TABLB OF LOGARITHMfC 






M. 


1 Sine 


1 ». 


1 Cosine 


1 D- 


I Tang. 


1 !>• 


1 Cotang. 


1 







9.808067 


251 


9.884254 


177 


9.923813 


427 


10.076187 


60 




1 


8082] 8 


251 


884148 


177 


924070 


427 


075930 


59 




2 


808368 


251 


884042 


177 


924327 


427 


075673 


58 




3 


808519 


250 


883933 


177 


92-1.583 


427 


075417 


57 




4 


808669 


250 


883829 


177 


924840 


427 


075160 


•56 




5 


808819 


2.50 


883723 


177 


925U96 


427 


074904 


55 




6 


808969 


250 


883617 


177 


925352 


427 


074648 


54 




7 


809119 


250 


883510 


177 


925609 


427 


074391 


53 




8 


809269 


250 


88:M04 


177 


9258;)5 


427 


074135 


52 




9 


809419 


249 


883297 


178 


926122 


427 


073878 


51 




10 


809569 


249 


883191 


178 


926378 


427 


073622 


50 




11 


9 809718 


249 


9.833084 


r8 


9.926634 


427 


10.073366 


49 




12 


809868 


249 


882977 


178 


92')890 


427 


073110 


48 




]3 


810017 


249 


882871 


178 


027147 


427 


072853 


47 




14 


810167 


249 


882764 


17S 


927403 


427 


072.197 


46 




15 


810316 


248 


882.>57 


178 


927659 


427 


072341 


45 




16 


810465 


248 


8d25.j0 


17S 


927915 


427 


072085 


44 




17 


810614 


248 


882443 


178 


923171 


427 


071829 


43 




18 


810763 


248 


882336 


179 


928427 


427 


071573 


42 




19 


810912 


248 


882229 


179 


928683 


427 


071317 


41 




20 


811061 


248 


882121 


179 


928940 


427 


071060 


40 




21 


9 811210 


248 


9.882014 


179 


9.929196 


427 


10.070804 


39 




22 


8 J 1358 


247 


881907 


179 


929452 


427 


070548 


38 




23 


81 1507 


247 


881799 


179 


929708 


427 


070292 


37 




24 


81 1655 


247 


881692 


179 


929964 


420 


070036 


36 




25 


811804 


247 


881584 


179 


930220 


426 


069780 


35 




26 


811952 


247 


881477 


17Q 


930475 


426 


01)9525 


34 




27 


812100 


247 


881369 


179 


930731 


426 


069269 


33 




28 


812248 


247 


881261 


180 


93)987 


426 


009013 


32 




29 


812395 


246 


881153 


180 


931243 


426 


068757 


31 




30 


812544 


246 


881046 


180 


931499 


426 


068501 


30 




31 


9 812692 


24^) 


9.880938 


180 


9.9317.55 


426 


10.068215 


29 




32 


812840 


246 


88U830 


130 


932010 


426 


067990 


28 




33 


812988 


210 


880722 


180 


932265 


426 


067734 


27 




34 


813135 


216 


880013 


180 


932522 


423 


067478 


26 




35 


813283 


216 


880505 


180 


932778 


425 


067222 


25 




36 


81 3430 


245 


880397 


180 


933033 


426 


066967 


24 




37 


813578 


2-15 


880239 


181 


933289 


426 


066711 


23 




38 


813725 


245 


880180 


181 


933545 


426 


066455 


22 




39 


813872 


245 


880072 


181 


933800 


423 


06()200 


21 




40 


814019 


245 


879963 


181 


934056 


426 


005944 


20 




41 


9.814166 


245 


9.879855 


181 


9.934311 


426 


10.065689 


19 




42 


814313 


245 


879746 


181 


934567 


426 


065433 


18 




43 


814460 


244 


879637 


181 


934823 


423 


065177 


17 




44 


814607 


244 


879529 


181 


935078 


426 


064922 


16 




45 


814753 


244 


879 '20 


181 


935333 


426 


064667 


15 




46 


81491)0 


244 


879311 


181 


935589 


426 


064411 


14 




47 


815046 


244 


879202 


182 


935844 


426 


064 156 


13 




48 


815193 


244 


879093 


182 


936100 


426 


063900 


12 




49 


815339 


244 


878984 


182 


936355 


426 


063645 


11 




50 


815485 


243 


878875 


182 


936610 


426 


063390 


10 




51 


9.815631 


243 


9.878760 


182 


9.936866 


425 


10.063134 







52 


815778 


243 


878656 


182 


937121 


425 


062879 


8 




53 


815924 


243 


878547 


182 


937370 


425 


062624 


7 




54 


816069 


243 


878438 


182 


937632 


425 


0623C»8 


6 




55 


816215 


243 


878328 


182 


937887 


425 


062113 


5 




56 


816361 


243 


878219 


183 


938142 


425 


061858 


4 




57 


816507 


242 


878109 


183 


938398 


425 


061t)()2 


3 




58 


816652 


242 


877999 


183 


938653 


425 


061347 


2 




59 


816798 


242 


877890 


183 


938908 


425 


061092 


1 




60 


816943 


242 


877780 


183 


939163 


425 


060837 









Cosine 




Sine 




1 CotHng. 


1 


1 Tang. 


M 





49 Degrees. 



SINES AND TANGENTS. (41 Degrees.) 



85 



M. 


1 Sine 


1 D 


i Cosine 


1 D. 


1 Tang. 


1 D- 


1 Cotang. 







9.8169-13 


242 


9.877780 


lfc'3 


y. 939] 63 


425 


10.060837 


60 


1 


817088 


242 


877G70 


183 


939418 


425 


060582 


59 


2 


817233 


242 


877.160 


183 


939673 


425 


060327 


58 


3 


817379 


242 


877450 


183 


939928 


425 


060072 


57 


4 


817524 


241 


877340 


183 


940183 


425 


059817 


56 


5 


817G68 


241 


877230 


184 


940438 


425 


0595G2 


55 


6 


817813 


241 


877120 


184 


940694 


425 


059306 


54 


7 


817958 


241 


877010 


184 


940949 


425 


059051 


53 


8 


818103 


241 


876899 


184 


941204 


425 


058796 


52 


9 


818247 


241 


876789 


184 


941453 


425 


058542 


51 


10 


818392 


241 


876678 


184 


941714 


425 


058286 


50 


11 


9.818536 


240 


9.876568 


184 


9.941968 


425 


10.058G32 


49 


12 


818G81 


240 


876457 


184 


942223 


425 


057777 


48 


13 


818325 


240 


876347 


184 


942478 


425 


057522 


47 


14 


8189G9 


240 


876236 


185 


942733 


42,5 


057267 


46 


15 


819113 


240 


876125 


185 


942988 


425 


057012 


45 


IG 


819257 


240 


876014 


185 


943243 


425 


05G757 


44 


17 


819401 


240 


875904 


185 


943498 


425 


05G5C2 


43 


18 


819545 


239 


875793 


185 


943752 


425 


056248 


42 


19 


819689 


239 


875C82 


185 


944007 


425 


055993 


41 


20 


819832 


239 


875571 


185 


9442G2 


425 


055738 


40 


21 


9.819976 


239 


9.875459 


185 


9.944517 


425 


10.05.5483 


39 


22 


820120 


239 


875348 


185 


944771 


424 


055229 


38 


23 


8202G3 


239 


875237 


185 


945026 


424 


054974 


37 


24 


820406 


239 


875126 


186 


945281 


424 


054719 


36 


25 


820550 


238 


875014 


186 


945535 


424 


0.544G5 


35 


2C 


820093 


238 


874903 


186 


945790 


424 


054210 


34 


27 


82083G 


238 


874791 


186 


946045 


424 


C53955 


33 


28 


820979 


238 


874680 


186 


946299 


424 


053701 


32 


29 


821122 


238 


874568 


186 


946554 


424 


053446 


31 


30 


821265 


238 


874456 


186 


946S08 


424 


053192 


30 


31 


9.821407 


238 


9.874344 


186 


9.947063 


424 


10.052937 


29 


32 


8215:10 


238 


8742.32 


187 


947318 


424 


052082 


28 


33 


821693 


237 


874121 


187 


947572 


424 


052428 


27 


34 


821835 


237 


874009 


187 


947826 


424 


052174 


26 


35 


821977 


237 


873896 


187 


948081 


424 


051919 


25 


36 


822120 


237 


873784 


187 


948,3.36 


424 


051. -64 


24 


37 


822262 


237 


873672 


187 


948590 


424 


051410 


23 


38 


822J04 


237 


873560 


187 


948844 


424 


051156 


22 


39 


822546 


237 


873448 


187 


9^9099 


424 


050901 


21 


40 


822688 


236 


873335 


187 


949353 


424 


050647 


20 


41 


9.822830 


236 


9.873223 


187 


9.949607 


424 


10.050393 


19 


42 


822972 


236 


873110 


188 


94'J8C2 


424 


050138 


18 


43 


823114 


236 


872998 


188 


950116 


424 


049S84 


17 


44 


823255 


236 


872585 


188 


950370 


424 


049630 


16 


45 


8^3397 


23G 


872772 


188 


950625 


424 


049375 


15 


46 


823539 


236 


872659 


188 


95U879 


424 


049121 


14 


47 


82.3-680 


236 


872547 


188 


951133 


424 


048867 


13 


48 


823821 


335 


872434 


188 


951388 


424 


048G12 


12 


49 


823963 


235 


87?321 


188 


951642 


424 


048358 


11 


50 


824104 


235 


872208 


188 


951896 


424 


048104 


10 


51 


9.824245 


235 


9.872095 


189 


9.952150 


424 


10.047850 


9 


52 


824386 


235 


871981 


189 


952405 


424 


047595 


8 


53 


824527 


235 


871808 


189 


952659 


424 


047341 


7 


54 


824668 


234 


871755 


189 


952913 


424 


047087 


6 


1 55 


824808 


234 


871641 


189 


953167 


423 


046833 


5 


56 


824949 


234 


871528 


189 


953421 


423 


040579 


4 


] 5"^ 


825090 


234 


871414 


189 


953675 


423 


046325 


3 


58 


82.5230 


234 


871301 


189 


953929 


423 


046071 


2 


59 


825371 


234 


871187 


189 


954183 


423 


045817 


1 


CO 


825511 


234 


871073 


190 


954437 


423 


045563 





1 


Cosine | 


1 


Sine 


I 


Cotang. 1 


1 


Tang. 1 


M. 



48 Degrees. 



86 


(42 Degrees.) a 


TABl.E OF LOGARITHMIC 




M. 


Sine 


1 D. 


I Cosine 


1 D. 


( Tang. 


1 »• 


1 Cotang 


ni 





9.8255U 


234 


9.871073 


190 


9.9.54437 


423 


10.045563 


60 


1 


825651 


233 


870960 


190 


954691 


423 


045309 


59 


2 


825791 


233 


870846 


190 


954945 


423 


045055 


58 


3 


825931 


233 


870732 


]90 


955200 


423 


044800 


57 


4 


820071 


233 


870618 


190 


95.5454 


423 


044546 


56 


5 


826211 


233 


870.504 


190 


955707 


423 


044293 


55 


6 


826351 


233 


870390 


190 


9.5.5961 


423 


044039 


54 


7 


826491 


233 


870276 


190 


956215 


423 


043785 


53 


8 


826run 


233 


870161 


190 


956469 


423 


043531 


52 


9 


826770 


232 


870047 


191 


956723 


423 


043277 


51 


10 


820910 


232 


869933 


191 


956977 


423 


043023 


50 


11 


9.827049 


232 


9.869818 


191 


9.957231 


423 


10.042769 


49 


12 


827189 


232 


869704 


191 


957485 


423 


042515 


48 


13 


827328 


232 


869.589 


191 


957739 


423 


042261 


47 


14 


827467 


232 


869474 


191 


957993 


423 


042007 


46 


15 


827606 


232 


869360 


191 


958246 


423 


041754 


45 


16 


827745 


232 


869245 


191 


958500 


423 


041500 


44 


17 


827884 


231 


869130 


191 


958754 


423 


041246 


43 


18 


828023 


231 


869015 


192 


959008 


423 


040992 


42 


19 


828162 


231 


868900 


192 


959262 


423 


040738 


41 


20 


828301 


231 


868785 


192 


959516 


423 


040484 


40 


21 


9.828439 


231 


9.868670 


192 


9.959769 


423 


10.040231 


39 


22 


828578 


231 


868555 


192 


96J023 


423 


039977 


38 


23 


828716 


231 


868440 


192 


960277 


423 


039723 


37 


24 


828855 


230 


868324 


192 


960.531 


423 


03Q469 


36 


25 


828D93 


230 


868209 


192 


960784 


423 


03J;:16 


35 


26 


829131 


230 


868093 


192 


96 J 038 


423 


038962 


34 


27 


829269 


230 


807978 


193 


961291 


423 


038709 


33 


28 


829407 


230 


867862 


193 


961.545 


423 


038455 


32 


29 


829545 


230 


867747 


193 


961799 


423 


038201 


31 


30 


829683 


230 


867631 


193 


962052 


423 


037948 


30 


31 


9.829821 


229 


9.867515 


193 


9.962306 


423 


10.037694 


29 


32 


829959 


229 


867399 


193 


962560 


423 


037440 


28 


33 


830097 


229 


867283 


193 


962813 


423 


037187 


27 


34 


830234 


229 


867167 


193 


963067 


423 


036933 


26 


35 


830372 


229 


867051 


193 


963320 


423 


036680 


25 


36 


830509 


229 


8G6935 


194 


963574 


423 


036426 


24 


37 


830646 


229 


8668 J 9 


194 


963827 


423 


036173 


23 


38 


830784 


229 


866703 


194 


964081 


423 


035919 


22 


39 


83U921 


228 


866586 


194 


964335 


423 


035665 


21 


40 


831058 


228 


866470 


194 


964588 


422 


035412 


20 


41 


9.831195 


228 


9.866353 


194 


9.964842 


422 


10.0.35158 


19 


42 


831332 


228 


866237 


194 


965095 


422 


034905 


18 


43 


831469 


228 


866120 


194 


965349 


422 


034651 


17 


44 


831606 


228 


866004 


195 


965602 


422 


034398 


16 


45 


831742 


228 


865887 


195 


965855 


422 


034145 


15 


46 


831879 


228 


865770 


195 


966109 


422 


033891 


14" 


47 


832015 


227 


865653 


195 


966362 


422 


03.3638 


13 


48 


832152 


227 


865536 


195 


966616 


422 


033384 


12 


49 


832288 


227 


865419 


195 


966869 


422 


033131 


11 


50 


832425 


227 


865302 


195 


967123 


422 


032877 


10 


51 


9.832561 


227 


9.865185 


195 


9.967376 


422 


10.032624 


9 


52 


832097 


227 


865(J68 


]95 


967629 


4-22 


032371 


8 


53 


8:^2833 


227 


864950 


195 


967883 


422 


032117 


7 


54 


832969 


226 


864833 


196 


968136 


422 


031864 


6 


55 


833105 


226 


864716 


196 


968389 


422 


031611 


^ i 


56 


833241 


226 


8(54598 


196 


968643 


422 


031357 


4 1 


57 


833377 


226 


864481 


196 


9l>889t> 


422 


031104 


3 


58 


833512 


226 


864363 


196 


969149 


422 


030851 


2 


59 


833648 


226 


864245 


196 


969403 


422 


030597 


1 


60 


833783 


226 


864127 


196 


969656 


4-22 


030344 







Cosine 


1 


Sine 1 




Cotang. 1 




Tang. 1 


M 



47 Degrees. 



SINES AND TANGENTS. (43 Degrecs.j 



87 



M. 


1 Sine 


1 B 


1 Cosine 


1 D. 


1 Tang. 


1 D. 


1 Cotang. 


1 





9.8;{;{7d3 


226 


9.864127 


193 


9.969650 


422 


10.030;;44 


GO 


1 


833919 


225 


864010 


196 


9699J9 


422 


0300'J1 


59 


2 


834054 


225 


863892 


197 


970162 


422 


029838 


58 


3 


834189 


225 


863774 


197 


970416 


422 


029584 


57 


4 


834325 


225 


863656 


197 


970()(i9 


422 


029331 


56 


5 


834460 


225 


8()3538 


197 


970J22 


422 


029078 


55 


6 


834595 


225 


8i)3419 


197 


971175 


422 


028325 


54 


7 


834730 


225 


863301 


197 


971429 


422 


0-28571 


53 


8 


834805 


225 


863183 


197 


971032 


422 


028318 


52 


9 


834999 


224 


863064 


197 


971935 


422 


028065 


51 


10 


835134 


224 


862946 


198 


972188 


422 


027812 


50 


11 


9.8352G9 


224 


9.862827 


198 


9.972441 


422 


10.0275.59 


49 


12 


835403 


224 


862709 


1J8 


972594 


422 


0273i;6 


48 


13 


835538 


224 


8!)2590 


198 


972918 


422 


027052 


47 


11 


835G72 


224 


862471 


198 


973291 


422 


02:)799 


46 


15 


835807 


224 


8G2353 


193 


973454 


422 


026546 


45 


l(i 


835941 


224 


862234 


198 


973707 


422 


02)293 


44 


17 


830J75 


223 


862115 


198 


97.i960 


422 


02!)040 


43 


18 


8362J9 


223 


861996 


193 


974213 


422 


02.5787 


42 


19 


836343 


223 


861877 


193 


9744i6 


422 


02.5534 


41 


20 


836477 


223 


861758 


199 


974719 


422 


025281 


40 


21 


9.836611 


223 


9.861638 


199 


9.974973 


422 


10.02.5027 


39 


1 22 


836745 


223 


861519 


199 


975226 


422 


024774 


38 


23 


836878 


223 


861400 


199 


975479 


422 


021521 


37 


24 


837012 


222 


861280 


199 


975732 


422 


024268 


36 


25 


837146 


222 


861161 


199 


975935 


422 


021015 


35 


2(5 


837279 


222 


861041 


199 


976238 


422 


023762, 


34 


27 


837412 


222 


860922 


199 


976491 


422 


023509 


33 . 


28 


837546 


222 


860802 


199 


976744 


422 


023250 


32 


29 


837679 


2.22 


860682 


203 


976997 


422 


023303 


31 


30 


837812 


222 


860562 


2j0 


977250 


422 


022750 


30 


31 


9.837945 


222 


9.860442 


200 


9.977.503 


422 


10.022497 


29 


32 


838078 


221 


8; '.0322 


200 


977756 


422 


022244 


28 


33 


838211 


221 


860202 


200 


9780 )9 


422 


021991 


27 


34 


838344 


221 


860082 


2C0 


978262 


422 


021738 


26 


35 


838477 


221 


859962 


200 


978515 


422 


021485 


25 


36 


838610 


221 


859342 


200 


978768 


422 


021232 


24 


37 


838742 


221 


8.)9721 


201 


979021 


4-22 


020979 


23 


38 


838375 


221 


859601 


201 


979274 


422 


020726 


22 


39 


839007 


221 


859480 


201 


979527 


422 


020473 


21 


40 


839140 


220 


859360 


201 


979780 


432 


020220 


2J 


41 


9.839272 


220 


9.859239 


201 


9.930033 


422 


10.019967 


19 


42 


839 104 


220 


859119 


201 


980236 


422 


019714 


18 


43 


839536 


220 


858933 


2)1 


980538 


422 


019462 


17 


44 


839r)68 


220 


85S877 


201 


930791 


421 


019209 


16 


45 


8.59800 


220 


8.>8756 


202 


931044 


421 


018956 


15 


46 


839932 


220 


858635 


202 


981297 


421 


018703 


14 


47 


840064 


219 


858514 


202 


931550 


421 


018450 


13 


48 


84019) 


219 


858393 


202 


981803 


421 


018197 


12 


49 


840328 


219 


358272 


202 


9820.56 


421 


017944 


11 


50 


840459 


239 


858151 


202 


982309 


421 


017691 


10 


51 


9.840591 


219 


9.8.58029 


2^2 


9.982.512 


421 


10.017438 


9 


52 


8407-22 


219 


857'.)08 


2.:)2 


982814 


421 


017186 


8 


53 


840854 


219 


857786 


202 


983067 


421 


016933 


7 


54 


840985 


219 


857665 


203 


983320 


421 


016680 


6 


55 


8411J6 


218 


8.57543 


203 


933573 


421 


016427 


5 


58 


841247 


218 


857422 


203 


983826 


421 


016174 


4 


57 


841378 


218 


857300 


203 


984079 


421 


015921 


3 


58 


841509 


218 


857178 


203 


984331 


421 


015669 


2 


59 


8416J0 


218 


857056 


203 


984584 


421 


01.5416 


1 


60 


841771 


218 


856934 


203 


984837 


421 


015163 







Cosiue 


1 


Sine 1 




Cotang. 




Tang. 1 


M. 



46 Degrees. 



88 




(44 Degrees.) log. sines and 


TANGENTS. 




M. 


1 Sine 


1 ». 


1 Cosine \ D. 


1 Tang. 


1 D. 


1 Cotang. 1 1 





9.841771 


218 


9.856934 


1 203 


9.9848.37 


421 


10.015163 


60 


1 


841902 


218 


856812 


I 203 


985090 


421 


0149J0 


59 


2 


842033 


218 


856690 


204 


985343 


421 


014657 


58 


3 


842163 


217 


850568 


204 


9S5596 


421 


014404 


57 


4 


842294 


217 


856446 


204 


985848 


421 


014152 


56 


5 


842424 


217 


856323 


204 


980101 


421 


013899 


55 


6 


842555 


217 


856201 


204 


9803.54 


421 


013646 


54 


7 


842685 


217 


856078 


204 


98GG07 


421 


01.3393 


53 


8 


8428 J 5 


217 


855956 


204 


9868G0 


421 


013140 


52 


9 


812946 


217 


855833 


204 


987112 


421 


012888 


51 


10 


&i3076 


217 


855711 


205 


987365 


421 


012635 


50 


11 


9.843206 


216 


9.855588 


205 


9.987618 


421 


10.012382 


49 


12 


843336 


216 


855465 


205 


987871 


421 


012129 


48 


13 


843466 


216 


855342 


205 


988123 


421 


011877 


47 


14 


843505 


216 


855219 


205 


988376 


421 


011624 


46 


15 


843725 


216 


855096 


205 


988629 


421 


011371 


45 


16 


843855 


216 


854973 


205 


988882 


421 


011118 


44 


17 


843984 


216 


854850 


205 


989134 


421 


C10806 


43 


18 


844114 


215 


854727 


206 


989387 


421 


010613 


42 


19 


844243 


215 


854603 


206 


989640 


421 


010360 


41 


20 


844372 


215 


854480 


206 


989893 


421 


010107 


40 


21 


9.844502 


215 


9.854356 


206 


9.990145 


421 


10.0098,55 


39 


22 


844631 


215 


854233 


206 


990398 


421 


009(J02 


38 


23 


844760 


215 


854109 


206 


990651 


421 


009349 


37 


24 


844889 


215 


853986 


206 


990903 


421 


009097 


36 


25 


845018 


215 


853862 


206 


991156 


421 


008844 


35 


26 


845147 


215 


853738 


206 


991409 


421 


008591 


31 


27 


845276 


214 


853614 


207 


991662 


421 


008338 


33 


28 


845405 


214 


853490 


207 


991914 


421 


00808(5 


32 


29 


845533 


214 


853366 


207 


992167 


421 


007833 


31 


30 


845662 


214 


853242 


207 


992420 


421 


007580 


30 


31 


9.845790 


214 


9.853118 


207 


9.992672 


421 


10 007328 


29 


32 


845919 


214 


852994 


207 


992925 


421 


007075 


28 


33 


846047 


214 


852869 


207 


993178 


421 


006822 


27 


34 


846175 


214 


852745 


207 


9934.30 


42] 


00(5570 


26 


35 


846304 


214 


852620 


207 


993683 


421 


006317 


25 


30 


846432 


213 


852496 


208 


993936 


421 


00(5064 


24 


37 


846560 


213 


852371 


208 


994189 


421 


005811 


23 


38 


846088 


213 


852247 


208 


994441 


421 


005559 


22 


39 


846816 


213 


852122 


208 


994694 


421 


005306 


21 


40 


846944 


213 


851997 


208 


994947 


421 


005053 


20 


41 


9.847071 


213 


9.851872 


208 


9-995199 


421 


10 004801 


19 


42 


847199 


213 


851747 


208 


9954.52 


421 


004.548 


18 


43 


847327 


213 


851622 


208 


995705 


421 


004295 


17 


44 


847454 


212 


851497 


209 


995957 


421 


004043 


16 


45 


847582 


212 


851372 


209 


996210 


421 


003790 


15 


46 


847709 


212 


851246 


209 


996463 


421 


003537 


14 


47 


847836 


212 


851121 


209 


996715 


42] 


003285 


13 


48 


847964 


212 


850996 


209 


996968 


421 


003032 


12 


49 


848091 


212 


850870 


209 


997221 


42J 


002779 


11 


50 


848218 


212 


850745 


209 


997473 


421 


002527 


10 


51 


9.848345 


212 


9.850619 


209 


9.997726 


421 


10.002274 


9 


52 


848472 


211 


850493 


210 


997979 


421 


002021 


8 


53 


848599 


211 


850368 


210 


998231 


421 


001769 


7 


54 


848726 


211 


850242 


210 


998484 


421 


001510 


6 


55 


848852 


211 


850116 


210 


998737 


421 


001263 


5 


56 


848979 


211 


849990 


210 


998989 


421 


001011 


4 


57 


849106 


211 


8498(54 


210 


999242 


421 


000758 


3 


58 


849232 


211 


849738 


210 


999495 


421 


000505 


2 


59 


849359 


211 


849611 


210 


999748 


421 


000253 


1 


60 


849485 


211 


849485 


210 


10.000000 


421 


000000 





1 


Cosine | 


1 


Sine 1 1 


Cotang. 1 


1 


Tang. 1 M. 1 



45 Degrees. 



TABLE 



OP 



NATUEAL SINES, COSINES, 
TANGENTS, AND COTANGENTS 



fOB 



EYEET DEGEEE AND MINUTE 

or THE QUADRANT. 



Note. — The minutes in the left-hand column of each page, increas- 
ing downwards, belong to the degrees at the top ; and those increasing 
upwards, in the right-hand column, belong to the degrees below. 



90 



NATURAL SINES. 





Deg. 


1 Deg. 


2 Deg. 


3 Deg. \ 


4 Deg. 




Nat. 


N.Co- 


Nat. iN. Co- 


Nat. 


N. Co- 


Nat. 


N. Co- 


Nat. 


N. Co- 





Sine 


Sine 


Sine. 


Sine 


Sine 
03490 


Sine 

99939 


Sine 
05^34 


SinA 
998( 3 


Sii>e 


Sine 


M 
60 


oaooo 


Unit. 


01745 


99J85 


06976 


99756 


1 


00029 


00000 


01774 


99934 


03519 


99938 


05263 


99361 


07605 


99754 


59 


2 


00053 


coooo 


01803 


99384 


03548 


99937 


05292 


99860 


07034 


99752 


58 


3 


00087 


00000 


01832 


99983 


o::577 


99930 


05321 


99*58 


07063 


99750 


57 


4 


00116 


00000 


018C2 


99383 


03006 


939,35 


0:350 


93857 


07092 


99748 


56 


5 


00145 


00000 


01891 


99982 


03635 


93334 


C53:9 


99355 


07121 


99746 


55 


6 


00175 


00000 


01920 


99382 


03664 


99933 


054J8 


9.854 


07150 


93744 


54 


7 


00204 


00300 


01949 


99981 


0.363.3 


93L32 


05437 


99352 


07179 


99742 


53 


8 


00233 


00000 


01978 


99380 


03723 


99931 


054;;6 


9b851 


07208 


<J9740 


52 


9 


00262 


00000 


02007 


99.383 


03752 


99930 


05495 


93849 


07237 


99738 


51 


10 


00291 


00000 


02036 


99379 


03781 


99923 


05524 


93347 


072(i6 


99736 


50 


Jl 


00320 


99999 


02065 


99379 


03810 


93327 


05553 


99843 


07295 


99734 


49 


]2 


00349 


99999 


02v;94 


99978 


03339 


93328 


05582 


99844 


07324 


99731 


48 


]3 


00378 


999J3 


02123 


93377 


03868 


99925 


G5G11 


93842 


07353 


99729 


47 


U 


00407 


99999 


02152 


99977 


03837 


93924 


05340 


93841 


07382 


99727 


46 


15 


00436 


99999 


02181 


99976 


03926 


99923 


05369 


99839 


07411 


99725 


45 


16 


00465 


99999 


02211 


99976 


03955 


99922 


05698 


99838 


07440 


99723 


44 


]7 


00495 


99999 


02240 


93975 


03384 


93921 


05727 


993:i6 


07469 


99721 


43 


18 


00524 


9999J 


02269 


99374 


04013 


93313 


05753 


99834 


07498 


99719 


42 


19 


00553 


99998 


02298 


99974 


04042 


93318 


05785 


99833 


07527 


99716 


41 


20 


00582 


99393 


02327 


99973 


04)71 


9.J917 


05814 


93831 


07556 


99714 


43 


21 


00611 


99998 


02356 


99372 


04100 


99313 


05841 


99823 


07585 


99712 


39 


22 


00G40 


93998 


02385 


93972 


04129 


03915 


05373 


99827 


07614 


99710 


38 


23 


006G9 


99398 


02414 


99971 


04159 


93313 


05902 


99826 


07643 


93708 


37 


24 


00638 


99998 


02443 


99370 


04188 


93912 


05331 


99824 


07672 


S9705 


36 


25 


00727 


99997 


02472 


93969 


042J7 


99311 


05360 


93822 


07701 


99703 


35 


2G 


00756 


93997 


02501 


93969 


04246 


93910 


05383 


99821 


07730 


93701 


34 


27 


00785 


99997 


02533 


99968 


04275 


93309 


0(:018 


99819 


07759 


93699 


33 


28 


00814 


99397 


02560 


99967 


04304 


93:)07 


06047 


93817 


07788 


99(;93 


32 


29 


00844 


99993 


02583 


99963 


04333 


99306 


06376 


99815 


07817 


99('94 


31 


30 


00873 


99996 


02618 


99968 


04362 


99905 


03105 


99313 


07846 


99G92 


30 


31 


00902 


99996 


02647 


99965 


04391 


99904 


06134 


99812 


07875 


99689 


29 


32 


00931 


99998 


02676 


99964 


04420 


93902 


03163 


99810 


07904 


99687 


28 


33 


00960 


99395 


02705 


93363 


04449 


93901 


03192 


99803 


07933 


99G85 


27 


34 


00989 


99995 


02734 


93963 


04478 


99900 


06221 


99803 


07962 


99683 


26 


35 


01018 


99995 


02703 


99962 


01507 


99838 


06250 


99804 


07991 


99680 


25 


36 


01047 


99335 


02792 


99361 


04536 


93337 


06279 


99833 


08020 


99678 


24 


37 


01076 


99934 


02821 


99960 


04565 


99896 


06308 


99801 


08049 


99676 


23 


38 


01105 


99994 


02850 


93959 


04594 


99894 


06337 


99799 


0S078 


99673 


2^2 


39 


01134 


99994 


02879 


99359 


04623 


99893 


06366 


99797 


08107 


99671 


21 


40 


01164 


99993 


02308 


99358 


04653 


99892 


06335 


99795 


08136 


99668 


20 


41 


01193 


99993 


02938 


99957 


04682 


99890 


06424 


99793 


08165 


99666 


19 


42 


01222 


99993 


02937 


93956 


04711 


99889 


064.33 


99732 


08194 


99664 


18 


43 


01251 


99932 


02993 


99355 


04740 


93888 


06482 


99790 


08223 


99661 


17 


44 


01280 


99392 


03025 


99:).54 


04769 


99886 


06511 


99788 


08252 


99659 


16 


45 


01309 


99991 


03054 


99953 


04798 


99885 


06540 


99786 


08281 


99657 


15 


46 


01338 


99991 


03083 


93952 


04827 


99883 


06569 


99784 


08310 


99654 


14 


47 


01367 


93391 


03112 


93352 


04856 


9!)832 


06598 


99782 


08339 


99652 


13 


48 


013i«j 


93990 


03141 


99951 


04885 


93881 


0(5627 


99780 


08368 


99649 


12 


49 


01425 


99990 


03170 


9935 ) 


04914 


99873 


06656 


99778 


08397 


99647 


11 


50 


01454 


99989 


03199 


99343 


04!>43 


99878 


06685 


99776 


08426) 


99644 


10 


51 


01483 1 


99989 


03228 


9^918 


04972 


9987^, 


06714 


99774 


084.15 


99642 


9 


52 


01513 


99989 


03257 


99947 


05001 


99875 


06743 


99772 


08484 


99639 


3 


53 


01542 


99988 


0328) 


9-3943 


05030 


99373 


06773 


99770 


08513 


99637 


7 


54 


01571 


99988 


03316 


93i)45 


05053 


99^72 


06802 


99768 


085 12 


99! '35 


6 


55 


01600 


99987 


03345 


99944 


05088 


99370 


06831 


99766 


08.571 


99('32 


5 


56 


01629 


99987 


03374 


93'.)43 


05117 


93869 


06860 


99764 


08600 


99630 


4 


57 


01()58 


999S6 


03403 


99942 


05146 


93867 


06889 


99762 


031)29 


99(527 


3 


58 


01687 


99986 


03432 


99911 


05175 


998n6 


06918 


99760 


08658 


99625 


2 


59 
M 


01716 


99985 

N S. 


03461 


99340 


05205 


99864 


0()947 


99758 


08687 


99622 


1 
M 


N.CS. 


N.CS. 


N.S. 


N. OS 


N.S. 


IV. cs. 


N.S. 


N CS 1 
85 J 


N 3. 
Deg. 


89 Deg. ' 


88 Deg. 1 


87 Deg. 1 


86 Deg. 1 



NATURAL SINES. 



91 



M 



5Deg. 


6 Deg. 


7 Deg. 


8 Deg. 1 9 Deg. 


M 
60 


N.S. 


N.CS. 

99619 


i\.S. 


\ cs 


N.S. (N.CS. 


^."sT 


N.CS. N.S. 


N. CS 


08716 


10453 


99452 


12id7 


992.55 


13917 


99027 15643 


98769 


I 


08745 


99617 


10482 


99449 


12216 


99251 


13946 


99023 


15672 


98764 


59 


2 


08774 


99614 


10511 


99446 


12245 


99248 


13975 


99019 


; 15701 


98760 


58 


3 


08803 


99612 


10540 


99443 


12274 


99J44 


14004 


99015 


1 15730 


98755 


57 


4 


08831 


99609 


10569 


99440 


12302 


99240 


14033 


99011 


! 15758 


98751 


56 


5 


08860 


99607 


10597 


99437 


12331 


99237 


14U61 


99006 


i 15787 


98746 


55 


6 


08889 


99604 


10626 


99434 


12360 


99233 


14090 


99002 


■ 15816 


98741 


54 


7 


08918 


99602 


10655 


99431 


123H1) 


99230 


14119 


9«998 


; 15845 


98737 


53 


8 


08947 


99599 


10684 


99428 


12418 


99226 


14148 


9d994 


1 15873 


98732 


52 


9 


08976 


99596 


10713 


99424 


12447 


99222 


14177 


98990 


15902 


98728 


51 


10 


09005 


99594 


10742 


99421 


12476 


99219 


14205 


98986 


15931 


98723 


50 


U 


09034 


99591 


10771 


99418 


12504 


99215 


14234 


98982 


15959 


98718 


49 


12 


09063 


99588 


10800 


99415 


12533 


99211 


14263 


98978 


15988 


98714 


48 


13 


0i)092 


99586 


10829 


99412 


12562 


99208 


14292 


98973 


16017 


98709 


47 


14 


09121 


99583 


10858 


99409 


12591 


99204 


14320 


98969 


16046 


98704 


46 


15 


U;)15i) 


99580 


10887 


99406 


12620 


99200 


14349 


98965 


16074 


98700 


45 


16 


09179 


99578 


10916 


99402 


12649 


99197 


14378 


98961 


16103 


98695 


44 


17 


09208 


99575 


10945 


99399 


12678 


99193 


14407 


98957 


16132 


98690 


43 


18 


09237 


99572 


10973 


99396 


12706 


99189 


14436 


98953 


1 16160 


98686 


42 


19 


09266 


99570 


110i)2 


99393 


12735 


99186 


14464 


98948 


i 16189 


98681 


41 


20 


092ij5 


99567 


11031 


99390 


12764 


99182 


14493 


98944 


16218 


98676 


40 


21 


oy324 


99564 


11000 


99386 


12793 


99173 


14522 


98940 


16246 


98671 


39 


22 


0ij353 


99562 


11089 


99383 


12822 


99175 


14551 


98936 


16275 


98667 


38 


23 


09382 


99559 


11118 


99380 


12851 


99171 


14580 


98931 


16304 


98662 


37 


24 


Od411 


99556 


11147 


99377 


12880 


99167 


14608 


98927 


16333 


98657 


36 


25 


09440 


99553 


11176 


99374 


12908 


99163 


14637 


98923 


16361 


98652 


35 


26 


0y469 


99551 


11205 


9J370 


12937 


90160 


14666 


98919 


16390 


98648 


34 


27 


09498 


99548 


11234 


99367 


12966 


99156 


14695 


98914 


1(>419 


98643 


33 


28 


09527 


99545 


11263 


99364 


12995 


99152 


14723 


98910 


16447 


99638 


32 


2^ 


09556 


99542 


11291 


99360 


13024 


99148 


14752 


98906 


16476 


98633 


31 


30 


09585 


99540 


11320 


99357 


13053 


99144 


14781 


98902 


16505 


98629 


30 


31 


09614 


99537 


11349 


99354 


13081 


99141 


14810 


98897 


16533 


98624 


29 


32 


09642 


99534 


11378 


99351 


13110 


99137 


14838 


98893 


16562 


98619 


28 


33 


09671 


99531 


11407 


99347 


13139 


99133 


14867 


98889 


16591 


98614 


27 


31 


09700 


99528 


11436 


99344 


13168 


99129 


14896 


98884 


16620 


98609 


26 


35 


09729 


99526 


11465 


99341 


13197 


99125 


14925 


98880 


16648 


98604 


25 


36 


09758 


99523 


11494 


99337 


13226 


99122 


14954 


98876 


16677 


93600 


24 


37 


09787 


99520 


11523 


99334 


13254 


99118 


14982 


98871 


16706 


98595 


23 


38 


09816 


99517 


11552 


99331 


13283 


99114 


15011 


98867 


16734 


98590 


22 


3d 


09845 


99514 


11580 


99327 


13312 


99110 


15040 


98863 


16763 


98585 


21 


40 


09874 


99511 


11609 


99324 


13341 


991i;6 


15069 


98858 


16792 


98580 


20 


41 


09903 


99508 


11638 


99320 


13370 


99102 


15097 


98854 


16820 


98575 


19 


42 


09932 


99506 


11667 


99317 


13399 


99098 


15126 


98849 


16849 


98570 


18 


43 


09961 


995J3 


11696 


99314 


13427 


99094 


15155 


98845 


16878 


98565 


17 


44 


09990 


99500 


11725 


99310 


13456 


9J091 


15184 


98841 


16906 


98561 


16 


45 


10019 


99497 


11754 


99307 


13485 


99087 


15212 


98836 1 


16935 


98556 


15 


46 


10048 


99494 


11783 


99303 


13514 


990S3 


15241 


98832 ■ 


16964 


98551 


14 


47 


10077 


9949 L 


11812 


99390 


13543 


99079 


15270 


98827 ; 


16992 


98546 


13 


48 


10106 


99488 


11840 


99297 


13572 


99075 


15292 


93823 


17021 


98541 


12 


49 


10135 


99485 


11869 


99293 


13600 


99071 


15327 


98818 


170.50 


98536 


11 


50 


10164 


99482 


11898 


99299 


13629 


99067 


15356 


98814 


17078 


88531 


10 


51 


10192 


99479 


11927 


99286 


13658 


990G3 


15385 


98809 ; 


17107 


98526 


9 


52 


10221 


99476 


11956 


99283 


13687 


99059 


15414 


98805 1 


17136 


98521 


8 


53 


10250 


99473 


11985 


99279 


13716 


99055 


15442 


98800 , 


17164 


98516 


7 


54 


10279 


99470 


12014 


99276 


13744 


99051 


15471 


98796 


17193 


98511 


6 


55 


10308 


99467 


12043 


99272 


13773 


99047 


15500 


98791 1 


17222 


98506 


5 


5G 


10337 


99464 


12071 


99269 


13>^02 


99043 


15.529 


98787 i 


17250 


9^501 


4 


57 


10366 


99461 


12100 


99255 


13831 


99039 


15557 


98782 ; 


17279 


98496 


3 


53 


10395 


99458 


12129 


992G2 


138B0 


99035 


15586 


93778 ' 


17308 


98491 


2 


59 
M 


10424 


99455 


12158 


99258 


13889 
N.CS 


99031 


15615 


98773 i 


17336 


98486 


1 

IT 


NCS 


N.S. 


N OS 


N S. 


N. S. 


N.CS 


N. S. , 


N.CS. 


N.S. 


84 Deg. 


83 Deg. J 


82 Deg. |j 


81 I 


)eg. 


80 Deg. 1 



92 



NATURAL SINES. 



M 



10 Beg. 


11 Beg. 


12 Beg. 


13 Beg. ij 14 Beg. 


M 
60 


N.S. 


N.CS. 
98481 


N.S. 
19081 


N.CS. 


N.S. 


N.CS. 
978 J 5 


N.S. 


N.CS. N.S. 


N.CS 


173G5 


98163 


20791 


22495 


97437 , 24192 


97030 


1 


17393 


98476 


19109 


98157 


20820 


97809 


22523 


97430 1 24220 


97023 


59 


2 


174-22 


98471 


19138 


98152 


20848 


97803 


225.5i» 


97424 1 24249 


97015 


58 


3 


17451 


98466 


19167 


98146 


20877 


97797 


2258( 


97417 24277 


97008 


57 


4 


17479 


98461 


19195 


93140 


20905 


97791 


22608 


97411 i 24305 


97001 


56 


5 


17508 


98455 


19224 


98135 


20933 


97784 


22637 


97404 i 243/3 


96994 


55 


6 


17537 


98450 


192.52 


98129 


20962 


97778 


22665 


97398 24302 


96987 


54 


7 


175G5 


98445 


19281 


98124 


20990 


97772 


22693 


97391 24390 


96980 


53 


8 


17594 


98440 


19.309 


98118 


21019 


97766 


22722 


97384 24418 


96973 


52 


9 


17G23 


98435 


19338 


98112 


21047 


97760 


22750 


97378 24446 


96966 


51 


10 


17651 


98430 


19366 


98107 


21076 


97754 


22778 


97371 124474 


96959 


50 


11 


17680 


98425 


19395 


98101 


21104 


97748 


22807 


97365 ; 24503 


96952 


49 


1-2 


17708 


98420 


19423 


98096 


21132 


97742 


22835 


97358 24531 


96945 


48 


13 


17737 


98414 


19452 


98090 


21161 


97735 


22863 


97351 24559 


96937 


47 


11 


17766 


98409 


19481 


98084 


21189 


97729 


22892 


97345 24587 


96930 


46 


15 


17794 


98404 


19509 


98079 


21218 


97723 


22920 


97338 24615 


96923 


45 


16 


17823 


98399 


19538 


98073 


21246 


97717 


22948 


97331 '24644 


96916 


44 


17 


17852 


98394 


19506 


98067 


21275 


97711 


22977 


97325 24672 


96909 


43 


18 


17880 


98389 


19595 


98061 


21303 


97705 


23005 


97318 24700 


90902 


42 


J9 


17909 


98383 


19623 


98056 


21331 


97698 


23033 


97311 24728 


96894 


41 


20 


17937 


98378 


196.52 


98050 


21360 


97692 


23062 


97304 24756 


96887 


40 


2-1 


17966 


98373 


19680 


98044 


21388 


97686 


23090 


97298 1 24784 


96880 


39 


22 


17995 


98368 


19709 


98039 


21417 


97680 


23118 


97291 124813 


96873 


38 


23 


18023 


98362 


197.37 


98033 


21445 


97673 


23146 


97284 124841 


96866 


37 


24 


18052 


98357 


19766 


98027 


21474 


97667 


23175 


97278 


24869 


96858 


36 


25 


J 3081 


983.52 


19794 


98021 


21502 


97661 


23203 


97271 


24897 


96851 


35 


26 


18109 


98347 


19823 


98016 


215.30 


97655 


23231 


97264 


24925 


96844 


34 


27 


18138 


98341 


19851 


98010 


21559 


97648 


23260 


97257 1 


24953 


96837 


33 


28 


18166 


98336 


19880 


98004 


21587 


97642 


23288 


97251 


24982 


96829 


32 


29 


18195 


98331 


19908 


97998 


21616 


97636 


23316 


97244 ; 


25010 


96822 


31 


30 


18224 


98325 


19937 


97992 


21644 


97630 


23345 


97237 


25038 


96815 


30 


31 


18252 


98320 


19965 


97987 


21672 


97623 


23373 


97230^ 


25066 


96807 


29 


32 


18281 


98315 


19994 


97981 


21701 


97617 


23401 ! 97223 ; 


25094 


96800 


28 


33 


18309 


98310 


20022 


97975 


21729 


97611 


23429 1 97217 i 


25122 


96793 


27 


34 


18338 


98304 


20051 


97969 


21758 


97604 


23458 ! 97210 I 


25151 


96786 


26 


35 


18367 


98299 


20079 


97963 


21786 


97598 


23486 i 97203 | 


25179 


96778 


25 


36 


18395 


98294 


20108 


97958 


21814 


97592 


2.3514 i 97196 


25207 


96771 


24 


37 


18424 


98288 


20136 


979.52 


21843 


97585 


23542 


97189 


25235 


96764 


23 


38 


18452 


98283 


20165 


97946 


21871 


97579 


23571 


97182 ; 


25263 


96756 


22 


39 


18481 


98-277 


20193 


97940 


21899 


97573 


23599 


97176 1 


25291 


96749 


21 


40 


18509 


93272 


20222 


97934 


21928 


97566 


23627 


97169 1 


25320 


96742 


20 


41 


18538 


9tS-267 


20250 


979-28 


219.56 


97560 


23656 


97162 1 


25348 


96734 


19 


42 


18567 


98261 


20279 


97922 


21985 


97553 


23684 


97155 1 


25376 


96727 


18 


43 


18595 


98256 


20307 


97916 


22013 


97547 


23712 


97148 i 


25404 


96719 


17 


44 


18324 


98-250 


20336 


97910 


22041 


97541 


23740 


97141 


25432 


96712 


16 


45 


18652 


98245 


20364 


97905 


22070 


97534 


23769 


97134 ! 


25460 


96705 


15 


46 


18681 


98240 


20393 


97899 


2-2098 


97528 


23797 


97127 


25488 


96697 


14 


47 


18710 


98-234 


20421 


97893 


22126 


97.521 


23825 


97120 i 


2551() 


96690 


13 


48 


18738 


98229 


204.50 


97887 


22155 


97515 


2:<853 


97113 1 


25545 


96682 


12 


49 


18767 


982-23 


20478 


97881 


22183 


97508 


2r.882 


97106 


25573 


96675 


11 


50 


18795 


98218 


20507 


97875 


2'2212 


97502 


2.3910 


97100 ! 


2.3601 


96667 


10 


51 


188-24 


98212 


20535 


97869 


2-2240 


97496 


23938 


97093 


25629 


96660 


9 


52 


18852 


98207 


20563 


97863 


22268 


97489 


23966 


97086 1 


25657 


96653 


8 


53 


18881 


98201 


20592 


97857 


22297 


97483 


23995 


97079 i 


25685 


96645 


7 


54 


18910 


98193 


20620 


97851 


22325 


97476 


24023 


97072 1 


25713 


96638 


6 


55 


18938 


98190 


20649 


97845 


22353 


97470 


24051 


970()5 


2.5741 


96630 


5 


56 


18967 


98185 


20()77 


97839 


22382 


974()3 


24079 


97058 


25769 


96623 


4 


57 


18995 


98179 


20706 


97833 


22410 


974.57 


24108 


97051 


25798 


96615 


3 


58 


19024 


98174 


207.34 


97827 


22438 


974.50 


24136 


97044 


25826 


96(H)8 


t) 


59 

m" 


19052 


98168 


20763 


97821 


22467 

N.CS. 


97444 


24164 


97037 


25854 

N CS 


9()f;00 

N S 


I 
M 




N.CS. 


N.S. 


N.CS. N.S. I' 


N.S. 


N.CS 


N.S. 


79 Beg. 


78Deg. I| 


77 Beg. 


7t) Beg. 1 


75 Deg. 1 



NATURAL SINKS. 



93 



M 



15 Deg. 


16 


Deg. 


17 Deg. 


18 Deg. 


1 19 


Deg. 


60 


N.S. 


N.CS. 
96593 


JN.S. 
27564 


N.CS. 


N.S. 


N.CS. 
95630 


N.S. 
30902 


N.CS. 


1 N.S. 


N. CS. 


25882 


96126 


29237 


95106 


32557 


94552 


1 


25910 


96585 


27592 


96118 


29265 


95622 


30029 


95097 


, 32.584 


94.542 


59 


o 


25938 


96578 


27620 


96110 


29293 


9.5613 


30957 


95088 


' 32612 


94533 


58 


3 


25966 


96570 


27648 


96102 


- 29321 


95605 


30985 


95079 


! 32639 


9452.3 


57 


4 


25994 


96562 


27676 


96094 


29348 


95596 


31012 


95070 


1 32667 


94514 


50 


5 


26022 


96555 


27704 


96086 


29376 


95588 


31040 


95061 


I 32694 


94504 


55 


6 


26050 


96547 


27731 


96078 


29404 


95579 


31068 


95052 


I 32722 


94495 


54 


7 


26079 


96540 


27759 


96070 


29432 


9.5571 


31095 


95043 


[ 32749 


94485 


53 


8 


26107 


96532 


27787 


96062 


29460 


95562 


31123 


95033 


1 32777 


94476 


52 


9 


26135 


96524 


27815 


96054 


29487 


95554 


31151 


95024 


32804 


94466 


51 


10 


26163 


96517 


27843 


96U46 


295 J 5 


95545 


31178 


95015 


^ 32832 


94457 


50 


U 


26191 


96509 


27871 


96037 


29543 


95536 


31206 


95006 


i 32859 


94447 


49 


12 


26219 


965U2 


27899 


96029 


29571 


95528 


31233 


94997 


1 32887 


94438 


48 


13 


26247 


96494 


27927 


96021 


29599 


95519 


31261 


94988 


1 32914 


94428 


47 


14 


26275 


96486 


27955 


96013 


29626 


95511 


31289 


94979 


1 32942 


94418 


46 


15 


26303 


96479 


27983 


00005 


29654 


95502 


31316 


94970 


' 32969 


94409 


45 


16 


26331 


96471 


28011 


95997 


29682 


95493 


31344 


94961 


' 32997 


94399 


44 


17 


26359 


96463 


28039 


95989 


29710 


95485 


31372 


94952 


33024 


94390 


43 


18 


26387 


96456 


28067 


95981 


29737 


95476 


31399 


94943 


33051 


94380 


42 


19 


26415 


96448 


28095 


95972 


29765 


95467 


31427 


94933 


33079 


94370 


41 


20 


26443 


96440 


28123 


95964 


29793 


95459 


31454 


94924 


33106 


94361 


40 


21 


26471 


96433 


28150 


95956 


29821 


95450 


31482 


94915 


33134 


94351 


39 


22 


26500 


96425 


28178 


95948 


29849 


9.5441 


31510 


94906 


33161 


94342 


38 


23 


26528 


96417 


28206 


95940 


29876 


9.5433 


31537 


94897 


33189 


94332 


37 


24 


26556 


96410 


28234 


95931 


29904 


95424 


31565 


94888 


33216 '94322 


36 


25 


26584 


96402 


2d262 


95923 


29932 


95415 


31593 


94878 


33244 


94313 


35 


26 


26612 


96394 


28290 


95915 


29960 


95407 


31620 


94869 


33271 


94303 


34 


27 


26640 


96386 


28318 


05907 


29987 


95398 


31648 


94860 


33298 


94293 


33 


28 


26668 


96379 


28346 


95898 


30015 


95389 


31675 


94851 


33326 


94284 


32 


29 


26696 


96371 


28374 


95800 


30043 


95380 


31703 


94842 


33353 


94274 


31 


30 


26724 


96363 


28402 


05882 


30071 


95372 


31730 


94832 


33381 


94264 


30 


31 


26752 


96355 


28429 


95874 


30098 


95363 


31758 


94823 


33408 


94254 


29 


32 


26780 


96347 


28457 


05865 


30126 


95354 


31786 


94814 


33436 


94245 


28 


33 


26808 


96340 


28485 


05857 


30154 


95345 


31813 


94805 


33463 


942.35 


27 


34 


26836 


96332 


28513 


05840 


30182 


95337 


31841 


94795 


33490 


94225 


26 


35 


26864 


96324 


28541 


95841 


30209 


95328 


31868 


94786 


3.3518 


94215 


25 


36 


26892 


96316 


28569 


95832 


30237 


95:U9 


31896 


94777 


33545 


94206 


24 


37 


26920 


96308 


28597 


95824 


30265 


95310 


31923 


94768 1 


33573 


94196 


23 


38 


26948 


96301 


28625 


95816 


30292 


95301 


31951 


94758 1 


33600 


94186 


22 


39 


26076 


96293 


28652 


95807 


30320 


95293 


31979 


94749 1 


33627 


94176 


21 


40 


27004 


96285 


28680 


95799 


30348 


95^284 


32006 


94740 j 


33655 


94167 


20 


41 


27032 


96277 


28708 


95791 


30376 


95275 


32034 


94730 1 


33682 


94157 


19 


42 


27060 


9()269 


28736 


95782 


30403 


95266 


32061 


94721 i 


33710 


94147 


18 


43 


27U88 


96261 


28764 


95774 


30431 


95257 


32089 


94712 ; 


33737 


94137 


17 


44 


27116 


96253 


28792 


95766 


30459 


95248 


32116 


94702 ; 


33764 


94127 


16 


45 


27144 


96246 


28820 


95757 


30488 


95240 


32144 


94693 


33792 


94118 


15 


46 


27172 


96238 


28847 


95749 


30514 


05231 


3217i 


94684 ! 


33819 


94108 


14 


47 


27200 


96230 


28875 


95740 


30542 


95222 


32199 


94674 


33846 


94098 


13 


48 


27228 


96222 


28903 


95732 


30570 


9.5213 


32227 


91 665 : 


33874 


94088 


12 


49 


27256 


9621 1 


28931 


95724 


30597 


95204 


32254 


94656 i 


33901 


94078 


M 


50 


27284 


96206 


28959 


95715 


30625 


95195 


32282 


94646 : 


33929 


94068 


10 


51 


27312 


96198 


28987 


95707 


30653 


95186 


32309 


94637 1 


33956 


94058 


9 


52 


27340 


9619U 


29015 


95698 


30680 


95177 


32337 


94627 1 


33983 


94049 


8 


53 


27368 


96182 


29042 


95690 


30708 


95168 


32364 


94618 1 


34011 


94039 


7 


54 


27396 


96174 


29970 


95631 


30736 


95159 


32392 


94609 ! 


34038 


94029 


6 


55 


27424 


96166 


29098 


95673 


30763 


95150 


32419 


94599 : 


34065 


94019 


5 


56 


27452 


96158 


29126 


95664 


30791 


95142 


32447 


94590 ; 


34093 


94009 


4 


57 


27480 


98150 


29154 


95656 


30819 


951.33 


32474 


94580 1 


34120 


93999 


3 


58 


27508 


96142 


29182 


95647 


30846 


95124 


32502 


94571 


34147 


93989 


2 


59 


27536 


96134 


29209 


95639 


30874 


95115 


32529 


94.561 i 


34175 


93979 


1 
M 


N.C^. 


N.S. 


N.CS. 


N.S. 


N.CS 


N.S. 


N.CS 


N.S. N.CSl 


N.S. 


74 Deg. 


73 Deg. 


72 Deg. 


71 Deg. i 70 Deg. 



94 



NATURAL SINES, 



M 



20 Deg. 


21 Deg. 


22 Deg. 


23 Deg. 


if 24 


Deg. 


n 


N.S. 


N.CS 
93U09 


35837 


N.Cs 


N.S. 
37401 


N.CS. 
92718 


IvTsT 


N.CS. 


iN.S. 


91355 160 1 


342U2 


93358 


39073 


92050 1 40674 


1 


34229 


93959 


35804 


93348 


37488 


92707 


39100 


92039 i 40700 


91343 


59 


2 


34257 


03949 


35891 


93337 


37515 


92097 


39127 


92028 40727 


91331 


58 


3 


34284 


93939 


35918 


93327 


37542 


92086 


39153 


92016 40753 


91319 


57 


4 


34311 


93929 


35945 


93310 


375(59 


92075 


39180 


92005 40780 


91307 


56 


5 


34339 


93919 


35973 


93300 


37595 


92064 


39207 


91994 40806 


91295 


55 


6 


34300 


93909 


30OO0 


93295 


37622 


92053 


39234 


91982 


40833 


91283 


54 


7 


34393 


93899 


3()027 


93285 


37049 


92042 


39200 


91971 


408G0 


91272 


53 


8 


34421 


93889 


30054 


93274 


37676 


92031 


39287 


91959 


40880 


91260 


52 


9 


34448 


93879 


30081 


93204 


37703 


92020 


39314 


91948 


40913 


91248 


51 


10 


34475 


93809 


30108 


93253 


37730 


92009 


39341 


91936 


40939 


91236 


50 


U 


34503 


93859 


30135 


93243 


37757 


92598 


39367 


91925 


40900 


91224 


49 


12 


34530 


93849 


36102 


93232 


37784 


92587 


39391 


91914 


40992 


91212 


48 


13 


34557 


93839 


30190 


93222 


37811 


92576 


39421 


91902 


141019 


91200 


47 


14 


34584 


93829 


30217 


93211 


37838 


92505 


39443 


91891 


41045 


91188 


46 


15 


34012 


93819 


30244 


932U 1 


37865 


92554 


39474 


91879 


41072 


91176 


45 


10 


34039 


93809 


30271 


93190 


37892 


92543 


39501 


91868 


41098 


91164 


44 


17 


34000 


93799 


30298 


93180 


37919 


92532 


39528 


91856 


141125 


91152 


43 


18 


34094 


93789 


30325 


93169 


37946 


92521 


39555 


91845 


41151 


91140 


42 


19 


34721 


93779 


30352 


93159 


37973 


92510 


39581 


91833 


41178 


91128 


41 


20 


34748 


93709 


30379 


93148 


37999 


92499 


:i.)60!^ 


91322 


41204 


91116 


40 


21 


34775 


93759 


30400 


93137 


38.)20 


92483 


39035 


91810 


41231 


91104 


39 


22 


34803 


93748 


30434 


93127 


38053 


92477 


39601 


91799 


41257 


91092 


38 


23 


34830 


93738 


30401 


93116 


38080 


92406 


39083 


91787 


41284 


91080 


37 


21 


34857 


93728 


30488 


9310() 


38107 


92455 


39715 


91775 


41310 


91008 


36 


25 


34884 


93718 


3h515 


93J95 


381.34 


92441 


3^j741 


91704 


41337 


91056 


35 


20 


34912 


93708 


33542 


9.>0d4 


38161 


92432 


39768 


91752 


413(J3 


91044 


34 


27 


34939 


93098 


30509 


93074 


38188 


92421 


39795 


91741 


41390 


91032 


33 


28 


34900 


93083 


30590 


93J03 


38215 


92110 


39822 


91729 


41416 


91020 


32 


29 


34993 


93077 


30023 


93052 


38241 


92399 


39343 


91718 


41443 


91008 


31 


30 


35021 


93007 


30050 


93042 


38268 


92388 


39875 


91706 , 


414(59 


90996 


30 


31 


35048 


93057 


30077 


93031 


38295 


92377 


39902 


91694 ' 


41496 


90984 


29 


32 


35075 


93047 


30704 


93020 


33322 


92306 


39928 


91083 


41522 


90972 


28 


33 


35102 


93037 


30731 


93010 


38349 


92355 


39955 


91671 


41549 


90900 


27 


34 


35130 


93020 


30758 


92J99 


38376 


92343 


39982 


91600 


41575 


90948 


26 


35 


35157 


93010 


30785 


92938 


33103 


92332 


40008 


91(548 


41602 


90930 


25 


30 


35183 


93600 


30812 


929; 8 


38430 


92321 


40035 


91036 


41628 


90924 


24 


37 


35211 


93590 


33839 


92907 


38450 


92310 


40002 


91625 


41655 


90911 


23 


38 


35239 


93585 


30807 


92950 


38483 


92299 


40083 


91613 


41681 


9089i) 


2^2 


39 


35200 


93575 


30894 


92945 


38510 


92287 


40115 


91601 


41?07 


90887 


21 


40 


35293 


93505 


30921 


92935 


38537 


92276 


40141 


91590 


41734 


90875 


20 


41 


35320 


93555 


36948 


92924 


38564 


92265 


40108 


91578 


41700 


90863 


19 


42 


35347 


93544 


3IJ975 


82913 


38591 


92254 


40195 


91.506 


41787 


90851 


18 


43 


35375 


93531 


37002 


92902 


38017 


92243 


40221 


91.5.55 


41813 


90839 


17 


44 


35402 


93524 


37029 


92392 


38644 


92231 


40248 


91543 


41840 


90826 


16 


45 


35429 


93514 


37056 


92881 


38071 


92220 


40275 


91531 


41866 


90814 


15 


46 


35450 


93503 


37083 


92870 


38698 


92209 


40301 


91519 


41892 


90802 


14 


47 


35484 


93493 


37110 


92859 


38725 


92198 


40328 


91.508 


41919 


90790 


13 


48 


35511 


93483 


37137 


92849 


33752 


92186 


40355 


91496 


41945 


90778 


12 


49 


35538 


93472 


37101 


92838 


38778 


92175 


40381 


91484 


41972 


90766 


11 


50 


35505 


93402 


37191 


92327 


38805 


92164 


40408 


91472 


41998 


90753 


10 


51 


35592 


93152 


37218 


92810 


.38332 


92152 


40434 


91461 i 


42024 


90741 


9 


52 


35019 


93441 


37215 


92805 


38859 


92141 


40461 


91449 


42051 


90729 


8 


53 


35047 


93431 


37272 


92794 


3888(5 


92130 


40488 


914.37 \ 


42077 


90717 


7 


54 


35074 


93420 


37299 


92784 


38912 


92119 


40514 


91425 ; 


42104 


90704 


6 


55 


35701 


93110 


37320 


92773 


38939 


92107 


40541 


91414 1 


42130 


90692 


5 


50 


35723 


93400 


37353 


92702 


38906 


92096 


40507 


91402 1 


42156 


90()80 


4 


57 


35755 


93389 


37380 


92751 


38993 


92085 


40594 


91390 i 


42183 


906(58 


3 


58 


35782 


93371) 


37407 


92740 


3<)()20 


92073 


40(521 


91378 


42209 


90055 


2 


59 


35810 


93308 


37434 


92729 


.39046 


92002 


40(547 


91.3(56 


42235 


90643 


1 
M 


N.CS. 


N.P. 


N.CS. 


N.S. 


N. CS. 


N.S. 


N.CS 


N.S. 


N.CS. 


*.S. 


Gi) Detr. 


68 Deg. 


67 Deg. 


H6 Deg. 


65 Deg. 1 



NATURAL SINES. 



95 





M 



25 


Deg. 


26 Deg. 


27 Deg. 


28 Deg. 1 29 Deg. 


1 
60 




N.S" 


~1V.CS 


. N.S. 
43837 


iV.CS 


N.S. 
45399 


N.CS 


N.sT 


N.CS 
88295 


. N.S. 


N.CS 




422(3\ 


> 90631 


80879 


fiiiivi 


46947 


48481 


87462 




1 


42-28^ 


^ 906 J 8 


43863 


898(i7 


45425 


89087 


46973 


88281 • 48506 


87448 


59 




o 


42315 


90606 


43889 


89854 


4.5451 


89074 


46999 


88267 48532 


87434 


58 




3 


42341 


90594 


43916 


89d4l 


45477 


8.0061 


47024 


88254 ! 48557 


87420 


57 




i 


423G7 


90582 


43942 


89828 


45.503 


89048 


47050 


88240 48583 


87406 


56 




5 


42.394 


90569 


43968 


89816 


45529 


89035 


47076 


88228 ^ 48)08 


87391 


55 







42420 


90557 


43994 


89803 


45554 


89021 


47101 


88213 ' 48634 


8/377 


54 




7 


42446 


90545 


44020 


89790 


4.3580 


89008 


47127 


88199 1 48659 


87363 


53 




8 


42473 


90532 


44046 


89777 


45606 


88995 


47153 


88185 [ 48684 


87349 


52 




9 


42499 


90520 


44072 


89764 


45032 


88981 


47178 


88172 


48710 


87335 


51 




in 


42525 


90507 


44098 


89752 


45658 


88968 


47204 


88158 


48735 


87321 


.50 




11 


42552 


90495 


44124 


89739 


45684 


88955 


47229 


88144 


48761 


87306 


49 




12 


42578 


90483 


44151 


89726 


45710 


88042 


47255 


88 J 30 


48786 


872y2 


48 




13 


42f)04 


90470 


44177 


89713 


45736 


88928 


47281 


88117 


48811 


87278 


47 




11 


42-01 


99458 


44203 


89700 


45762 


88915 


47306 


88103 1 48837 


87264 


46 




15 


42657 


90446 


44229 


89687 


45787 


88902 


47332 


88089 1 48862 


8? 250 


45 




1(3 


42683 


90433 


44255 


89674 


45813 


88888 


47358 


88075 ! 48888 


87235 


44 




17 


42709 


90421 


44281 


80662 


45839 


88875 


47383 


88062 1 48913 


87221 


43 




18 


42736 


90408 


44307 


89649 


45865 


88862 


47409 


88048 i 48938 


87207 


42 




19 


42762 


90396 


41333 


89636 


4.5891 


88848 


47434 


8S034 ij 48964 


87193 


41 




20 


42788 


90383 


44359 


89623 


45917 


88835 


47460 


8:020 1 148989 


87178 


40 




21 


42815 


90371 


44385 


89610 


45942 


88822 


47486 


88006 ! 49014 


87164 


39 




22 


42841 


90358 


44411 


89597 


45968 


88808 


47511 


87993 


4>0040 


87150 


38 




23 


42867 


90346 


44437 


89584 


45994 


88795 


47537 


87979 


49065 


87136 


37 




24 


42894 


90334 


44464 


89571 


46020 


88782 


47562 


87965 


49090 


87121 


36 




25 


42920 


90321 


44490 


89558 


46046 


88768 


47588 


87951 


49116 


87107. 


35 




2{i 


42946 


90309 


44516 


89545 


46072 


88755 


47614 


87937 


49141 


87093 


34 




27 


42972 


90296 


44542 


80532 


46097 


88741 


47C39 


87923 


49166 


87079 


33 




28 


42999 


90284 


44568 


89519 


46123 


88728 


47665 


87909 


49192 


87064 


32 




29 


43025 


90271 


44594 


89506 


46J49 


83715 


47690 


87896 


40217 


87050 


31 




30 


43051 


90259 


44620 


89493 


4G175 


88701 


47716 


87882 


49242 


87036 


30 




31 


43077 


90246 


44646 


89480 


46201 


88688 


47741 


878G8 


49268 


87021 


29 




32 


43104 


90233 


44672 


80467 


46226 


88674 


47767 


87854 


49293 


87007 


28 




33 


43130 


90221 


44698 


89154 


46252 


88661 


47793 


87840 


49318 


86993 


27 




34 


43156 


90208 


44724 


89441 


46278 


88(547 


47818 


87826 ' 


49344 


8G978 


26 




35 


43182 


90196 


44750 


89428 


46304 


88634 


47844 


87812 


49369 


8o9G4 


25 




3G 


43209 


90183 


44776 


89415 


46330 


88620 


47869 


87798 


49394 


86949 


24 




37 


43235 


90171 


44802 


89402 


46355 


88607 


47895 


87784 ^ 


49419 


80935 


23 




38 


43261 


90158 


44828 


89389 


46381 


88593 


47920 


87770 


49445 


86921 


22 




39 


43287 


90146 


44854 


89376 


46407 


88580 


47946 


87756 : 


49470 


85906 


21 




40 


43313 


90133 


44880 


89363 


46433 


88566 


47971 


87743 


49495 


8o892 


20 




41 


43340 


90120 


44906 


89350 


46458 


88553 


47997 


87729 : 


49521 


86878 


19 




42 


43366 


90108 


44932 


89337 


46484 


83539 


48022 


87715 \ 


49.546 


86863 


18 




43 


43392 


90095 


44958 


89324 


46510 


88526 


48048 


87701 ! 


49571 


86849 


17 




44 


43418 


90082 


44984 


80311 


46536 


88512 


48073 


87687 j 


49596 


86834 


16 




45 


43445 


90070 


45010 


89298 


46561 


88499 


48099 


87673 1 


49622 


8C820 


15 




4G 


43471 


90057 


45036 


89285 


46587 


88485 


48124 


87659 ' 


49647 


86805 


14 




47 


43497 


90045 


45062 


80272 


46613 


88472 


48150 


87645 ' 


49672 


86791 


13 




48 


43523 


90032 


45088 


80259 


46639 


88458 


48175 


87631 ' 


49r97 


80777 


12 




49 


43549 


90019 


45114 


89245 


46664 


88445 


48201 


87617 1 


49723 


86762 


11 




50 


43575 


90907 


45140 


89232 


46690 


88431 


48226 


87603 j 


49748 


86748 


10 




51 


43G02 


89994 


45166 


89219 


46716 


88417 


48252 


87589 


49773 


86733 


9 




52 


43628 


80931 


45192 


89206 


46742 


88404 


48277 


87575 1 


49798 


86719 


8 




53 


43654 


89968 


45218 


89193 


46767 


88390 


48303 


87561 i 


49824 


86704 


7 




54 


43C89 


8905G 


45213 


89180 


46793 


88377 


48328 


87546 ' 


49849 


86690 


6 




55 


43706 


80943 


45269 


89167 


46819 


88363 


48354 


87532 i 


49874 


86675 


5 




56 


43733 


89930 


45295 


89153 


46844 


88349 


48379 


87518 1 


49899 


86661 


4 




57 


43759 


80018 


45321 


80140 


4G870 


88336 


48405 


87504 ! 


49924 


86646 


3 




58 


43785 


89905 


45347 


89127 


46896 


88322 


48430 


87490 j 


49950 


86632 


c> 




59 


43811 


89802 


45373 


89114 


46921 


88308 


48456 


87476 


49975 


86617 


1 


i 


N CS. 


N__P. 


iV.CS. 


N.S. 


N. CS. 


N.S. 


N.CS 


N.S. 


N.CS 


N.S. 


1 

1 


64 Deg. 


63 Deg. II 


62 Deg. II 


61 Deg. 1 


60 Deg. 1 



96 



NATURAL SINES. 



M 


30 Deg. 


31 Deg. 


32 Deg. 


33 Deg. II 34 Deg. 


M 
60 


nTs: 


N.CS. 


JN.S. 


N.CS 


N.S. 
52992 


N.CS. 


N.sT 


N.CS. N.S. 
83867 55919 


N.CS 





50000 


866.i3 


51504 


85717 


848u5 


54464 


82904 


1 


50025 


86588 


51529 


85702 


53017 


84789 


54488 


83851 


55U43 


82887 


59 


2 


50050 


86573 


51554 


F5687 


53041 


84774 


54513 


83835 


1 559f'8 


82871 


58 


3 


50076 


83559 


51579 


85672 


53066 


84759 


54537 


83819 


55992 


82855 


57 


4 


50101 


86544 


51604 


85657 


53091 


84743 


54561 


83804 


' 56016 


82839 


56 


5 


59126 


86530 


51628 


85642 


53115 


84728 


54.586 


83788 


: 56040 


82822 


55 


6 


50]51 


86515 


51653 


85627 


53140 


84712 


54610 


83772 


1 56004 


82806 


54 


7 


50176 


86501 


51678 


85612 


53164 


84697 


54635 


83756 


56088 


82790 


53 


8 


50201 


8G486 


51703 


8.5597 


53189 


84681 


54659 


83740 


1 56112 


82773 


52 


9 


50227 


86471 


51728 


85582 


53214 


84666 


54683 


83724 


56136 


82757 


51 


10 


50252 


80457 


51753 


85567 


53238 


84650 


54708 


83708 


56160 


82741 


50 


11 


50277 


8G442 


51778 


85551 


53263 


84635 


54732 


83692 


56184 


82724 


49 


12 


50302 


86427 


51803 


85536 


53288 


84619 


54756 


83676 


56208 


82708 


48 


13 


50327 


86413 


51828 


85521 


53312 


84604 


54781 


83660 


56232 


82692 


47 


14 


50352 


86398 


51852 


8550G 


53337 


84588 


54805 


83645 


56256 


82675 


46 


15 


50377 


86384 


51877 


85491 


53361 


84573 


54829 


83629 


56280 


82659 


45 


IG 


50403 


86369 


51902 


85476 


53386 


84557 


54854 


83613 


56305 


82643 


44 


17 


50428 


86354 


51927 


85461 


53411 


84542 


54878 


83597 


56329 


82626 


43 


Irt 


50453 


8G340 


51952 


8.5446 


53435 


84526 


54902 


83581 


56353 


82610 


42 


19 


50478 


86325 


51977 


85431 


53460 


84511 


54927 


83565 


56377 


82593 


41 


20 


50503 


88310 


52002 


8.5416 


53484 


84495 


54951 


83549 


56401 


82577 


40 


21 


50528 


86295 


5202G 


85401 


5^509 


84480 


54975 


83533 


56425 


82.561 


39 


22 


50553 


86281 


52051 


85385 


53534 


844(]4 


54999 


83517 


56449 


82544 


38 


23 


50578 


86266 


52076 


85370 


53558 


84448 


55024 


83.501 


56473 


82528 


37 


24 


50603 


8C251 


52101 


85355 


53583 


84433 


5.5018 


83485 


56497 


8251 1 


36 


25 


50628 


86237 


.52126 


8.5340 


53607 


84417 


5,5072 


83469 


56521 


82495 


35 


26 


50654 


86222 


52151 


85325 


53632 


84402 


55097 


83453 


56545 


82478 


34 


27 


50f379 


86207 


52175 


85310 


53656 


84383 


55121 


83437 


56569 


82462 


33 


28 


50704 


86192 


52200 


85294 


53f-81 


84370 


.55145 


83421 


56593 


82446 


32 


29 


50729 


8G178 


52225 


85279 


5.3705 


843.55 


.55169 


83405 


56617 


82429 


31 


30 


50754 


86163 


52250 


85264 


53730 


84339 


55194 


83389 


56641 


82413 


30 


31 


50779 


86148 


52275 


85249 


53754 


84324 


5.5218 


83373 


56665 


82396 


29 


32 


50804 


86] 33 


52299 


85234 


53779 


84308 


5.5242 


83356 


56689 


82:i80 


28 


33 


508-29 


8G1J9 


52324 


8.5218 


53804 


84292 


55266 


83340 


56713 


82363 


27 


34 


50854 


86104 


52349 


85203 


53828 


84277 


55291 


83324 


56736 


82347 


20 


35 


50879 


86089 


52374 


85188 


53853 


84261 


5.5315 


83308 


56760 


82330 


25 


36 


50904 


86074 


52399 


85173 


53877 


84245 


55339 


83292 


56784 


82314 


24 


37 


50929 


86059 


52423 


85157 


53902 


84230 


5.5363 


83276 


56808 


82297 


23 


38 


50954 


86045 


52448 


85142 


53920 


84214 


55388 


83260 


56832 


82281 


22 


39 


50979 


86030 


52473 


85127 


5.3951 


84198 


.55412 


83244 : 


56856 


82264 


21 


40 


51004 


860 J 5 


52498 


85112 


53975 


84182 


55436 


83228 


56880 


82248 


20 


41 


51029 


86000 


52522 


85096 


54000 


84167 


55460 


8.3212 


56904 


82231 


19 


42 


51054 


85935 


52547 


85081 


54024 


84151 


5.5484 


83195 


56928 


82214 


IS 


43 


51079 


85970 


52572 


85066 


54049 


84135 


.5.5509 


83179 


56952 


82198 


17 


44 


51104 


85956 


52597 


8.5051 


54073 


84120 


55533 


83163 


56976 


82181 


16 


45 


51129 


85941 


52621 


85035 


54097 


84104 


55557 


83147 ^ 


57000 


82165 


15 


46 


51154 


85926 


52646 


85020 


54122 


84088 


55581 


83131 i 


57024 


82148 


14 


47 


51179 


85911 


52671 


8.5005 


.54146 


84072 


55005 


83115 


57047 


82132 


13 


48 


51204 


85893 


52(;93 


84989 


54171 


84057 


.55630 


83098 


57071 


82115 


12 


49 


51229 


85881 


52720 


84974 


54195 


84041 


556.54 


83082 


57095 


82098 


11 


50 


51254 


85866 


52745 


84959 


54220 


84025 


55678 


83066 


57119 


82082 


10 


51 


51279 


85851 


52770 


84943 


.54244 


84009 


55702 


83050 i 


57143 


82065 


fl 


52 


51304 


85836 


52794 


84928 


54269 


83994 


5.5726 


83034 1 


57167 


82048 


8 


53 


51329 


85821 


52819 


84913 


54293 


8.3978 


55750 


8.3017 i 


57191 


82032 


7 


54 


51354 


85806 


52844 


84897 


.54317 


83962 


55775 


83001 1 


57215 


82015 


6 


55 


51379 


85792 


52869 


84882 


54342 


83946 


55799 


82985 ' 


57238 


81999 


5 


5G 


51404 


85777 


52893 


84866 


54366 


83930 


55823 


82969 : 


57262 


81982 


4 


57 


51429 


85762 


52918 


84851 


54391 


83915 


55847 


82953 1 


57280 


81965 


3 


58 


51454 


85747 


52913 


84836 


54415 


83899 


5.5871 


82936 ! 


57310 


81949 


o 


59 


51479 


85732 


529!)7 


84820 


54440 


83883 


55895 


82920 ; 


57334 


81932 


1 


N.Cri. 


N. S. 


N.CS. 


N.S. 


N.CS 


N.S. 


N.CS 


N.S. 1 


N.CS 


N S. 


59 Deg. 


58 Deg. 


57 1 


)eg. 


56 Deg. 


55 Deg. 



NATURAL SINES. 



97 








35 


D^g. 


36 Deg. 


37 Deg. 


38 Deg. . 39 Deg. 


' 60 




N.s: 


~Kcs 


N.S. 


N.CS 


N.S, 


N.CS. 

I 79864 


' N.s: 


N.CS. N.S. 
78801 62932 


N.CS 
' 777"l5 




5735t 


81915 


58779 


80902 


6018S 


61561 




1 


57381 


81899 


58802 


80885 


60205 


711846 


61589 


78783 62955 


77696 


59 




o 


57405 


81882 


58826 


80867 


6022c 


79829 


61612 


78765 62977 


77678 


58 




3 


57429 


81865 


58849 


80850 


60251 


79811 


61635 


78747 63000 


77(60 


57 




4 


57453 


81848 


58873 


80833 


60274 


79793 


61658 


78729 63022 


77641 


56 




5 


57477 


81832 


58896 


80816 


60298 


79776 


61681 


78711 63015 


77623 


55 




G 


57501 


81815 


58920 


80799 


60321 


79758 


61704 


78694 63068 


771505 


54 




7 


57524 


81798 


58943 


80782 


69344 


79741 


61726 


78670 63090 


7:586 


53 




8 


57548 


81782 


58967 


80765 


60367 


79723 


61749 


78658 63113 


77568 


52 




9 


57572 


81765 


58990 


80748 


60390 


7970(5 


61772 


78610 63135 


77550 


51 




10 


57596 


81748 


59014 


80730 


60414 


796S8 


61795 


78622 63158 


77531 


50 




11 


57619 


81731 


59;)37 


80713 


60437 


79371 


61818 


78604 63183 


77513 


49 




1-2 


57643 


81714 


59061 


80696 


60460 


79653 


61841 


7858;3 63203 


77494 


48 




13 


57667 


81698 


59084 


80679 


60483 


79635 


(518(34 


7.^568 63225 


77476 


47 




14 


57691 


81681 


59108 


80IJ62 


60506 


79618 


61887 


78550 63248 


77^58 


46 




15 


57715 


81664 


59131 


8J644 


60529 


79600 


61909 


78532 63271 


77439 


45 




16 


57738 


81647 


59154 


89627 


60553 


79533 


C1932 


78514 


; 63293 


77421 


44 




17 


57762 


81631 


59178 


80G10 


60576 


79535 


61955 


78490 


1 63316 


77402 


43 




18 


57786 


81614 


59201 


81)593 


60599 


79517 


61978 


78478 


i 63338 


77384 


42 




19 


57810 


81597 


59225 


80576 


6(;622 


79530 


62001 


784fi0 


63361 


77;.6ii 


41 




20 


57833 


81580 


59248 


80558 


60645 


79512 


62024 


78442 63383 


77347 


40 




21 


57857 


81563 


59272 


80541 


60668 


79494 


(32046 


78424 63406 


77329 


39 




22 


57881 


81546 


59295 


80524 


60691 


79477 


620f59 


78405 63428 


77310 


38 




23 


57904 


8 J 530 


59318 


80507 


(30714 


79459 


62392 


78387 63451 


77292 


37 




24 


57928 


81513 


59342 


80489 


60738 


79141 


62115 


78:369 63473 


77273 


36 




25 


57952 


81496 


59365 


80472 


60761 


79424 


621.38 


78351 , 63496 


77255 


35 




20 


57976 


81479 


59389 


83455 


60784 


79406 


(321(50 


78333 


; 63518 


77230 


34 




27 


57999 


81462 


59412 


80438 


608J7 


79388 


02183 


78315 


, 63540 


77218 


33 




28 


58023 


81445 


59436 


80420 


60830 


79371 


(5220! 5 


78297 


63563 


77199 


32 




29 


58047 


81423 


59459 


80403 


60853 


79353 


62229 


78279 


63.385 


77181 


31 




30 


58070 


81412 


59482 


80386 


60876 


79335 


62251 


78261 


63608 


77102 


30 




31 


58094 


81395 


59506 


80368 


60899 


79318 


62274 


78243 


03630 


77144 


29 




32 


58118 


81378 


59529 


80351 


60922 


79300 


62297 


78225' 


63653 


77125 


28 




33 


58141 


81361 


59552 


80334 


60945 


79282 


62320 


78200 


63675 


77107 


27 




34 


58165 


81344 


59576 


8;J316 


60968 


79234 


62342 


78188 


63698 


77088 


26 




35 


58 J 89 


81327 


59599 


80299 


60991 


79247 


62365 


78170 


63720 


77070 


25 




36 


58212 


81310 


59(522 


80282 


61015 


79229 


62388 


78152 


03742 


77051 


24 




37 


58236 


81293 


50646 


80264 


61033 


79211 


6241 1 


78134 037(55 


77033 


23 




38 


58260 


81276 


59669 


80247 


61061 


79 J 93 


62433 


78110 63787 


77014 


22 




39 


58283 


81259 


59693 


80230 


61084 


79176 


(52456 


78098 63810 


76996 


21 




40 


58307 


81242 


59716 


80212 


61107 


79158 


62479 


78079 


63832 


76977 


20 




41 


58330 


81225 


59739 


80195 


61130 


79140 


62502 


78061 


63854 


76959 


19 




42 


58354 


81208 


59763 


80178 


61153 


79122 


62524 


78043 


63877 


76940 


18 




43 


58378 


81191 


59786 


801(50 


61176 


79105 


62547 


78025 


63899 


75921 


17 




44 


58401 


81174 


59809 


80143 


61199 


79087 


62570 


78007 


63922 


76903 


16 




45 


5S425 


81157 


59832 


80125 


61222 


79069 


62592 


77988 


63944 


76884 


15 




46 


58449 


81140 


59856 


80108 


61245 


79051 


(52615 


77970 


63966 


76866 


14 




47 


58472 


81123 


59879 


89091 


61268 


79033 


62638 


77952 


63989 


76847 


13 




48 


5849G 


81106 


59902 


80073 


61291 


79015 


62(560 


77934 


64011 


76828 


12 




49 


58519 


81089 


59926 


80056 


61314 


78998 


62(383 


77916 


64033 


76810 


11 




50 


58543 


81072 


59949 


80038 


61337 


78980 


62706 


77897 


64056 


76791 


10 




51 


58567 


81055 


59972 


80021 i 


613(30 


789(32 


(52728 


77879 


64078 


76772 


9 




52 


58590 


81038 


59995 


80003 1 


61383 


78944 


62751 


7-861 


64100 


76754 


8 




53 


58614 


81021 


60019 


79986 


61406 


78926 


62774 


77843 


64123 


76735 


7 




54 


58637 


81004 


60042 


79968 


6 J 429 


78908 1 


627'96 


77824 


64145 


76717 


6 




55 


58661 


80987 


600(55 


79951 


61451 


78891 i 


62819 


77806 


64167 


76698 


5 




56 


58684 


80970 


60089 


79934 


61474 


78873 


62842 


77788 


64190 


76679 


4 




57 


58708 


80953 


60112 


799 J 6 


61497 


78855 


62864 


77769 


64212 


76661 


3 




58 


58731 


80936 


(50135 


79S99 


61520 


78837 


(52887 


77751 i 


64234 


76642 


2 




59 


58755 


80919 I 

N. S. 1 


60158 


79881 


61543 


78819 


62909 


77733 


6425(5 

N CS. 


76623 

N S. 


1 
M 

J 




N.CS. 


N. CS. 


N.S. 


N. CS 


N.S. 


N. CS 


N. S. 




54 Deg. 


53 Deg. 1 


52 Deg. II 


51 Deg. 1 


50 Deg. 



ellwood's test prob. — 7 



98 



NATURAL SINKS. 





M 




40 Deg. 


41 Deg. 


42 Deg. 


43 Deg. 


44 Deg. 






N. S. 


N.CS. 


N.S. 
656U6 


N.CS. 


N.S. 


N.CS. 
74314 


N.S. 


jN.CS. 


N^.'sT 

69466 


\. CS. 


M 




64279 


76604 


75471 


66913 


68200 


73135 


71934 60 j 




1 


64301 


76586 


6562S 


7.5453 


66935 


74295 


68221 


73116 


69487 


71914 59 1 




o 


64323 


76567 


65650 


75433 


66956 


74276 


68242 


73(i96 


69.508 


71894 


58 




3 


64346 


76548 


65672 


7.5414 


60978 


74256 


G8264 


73.i76 


69.529 


71873 


57 




4 


64368 


76530 


65694 


75395 


6i999 


74237 


68285 


73056 


69549 


71853 


56 




5 


64390 


76511 


65716 


75375 


67021 


74217 


68366 


73036 


69570 


71833 


55 




G 


64412 


76492 


65738 


75356 


67043 


74198 


68327 


73016 


69.591 


71813 


54 




7 


64435 


76473 


65759 


75337 


67064 


74178 


68349 


729b6 


69(512 


71792 


53 




8 


64457 


76455 


6.5781 


75318 


67086 


74 J. 59 


68370 


72976 


69633 


71772 


52 




9 


64479 


76436 


65803 


75299 


67107 


74 J 39 


68391 


72957 


(59654 


71752 


51 




10 


64501 


76417 


65825 


75280 


67129 


74120 


68412 


72937 


69675 


71732 


50 




IJ 


64524 


76398 


65847 


75261 


67151 


741U0 


68433 


72917 


69G96 


71711 


49 




12 


64546 


76380 


65869 


75241 


67172 


74080 


68455 


728S-)7 


69717 


71691 


48 




13 


64568 


76361 


65891 


75222 


67 J 94 


74061 


68476 


72877 


69737 


71671 


47 




14 


64590 


76342 


60-913 


75203 


67215 


74041 


68497 


72857 


69758 


71650 


46 




15 


64612 


76323 


65935 


75184 


67237 


74022 


68518 


72837 


69779 


71630 


45 




16 


64635 


76304 


65956 


75165 


67258 


74002 


68539 


72817 


69303 


71610 


44 




17 


64657 


76286 


65978 


75146 


67280 


73983 


6856 1 


72797 


69821 


71590 


43 




18 


64679 


76267 


6S000 


75126 


67301 


739G3 


68582 


72777 


69842 


71.569 


42 




19 


64701 


76248 


66022 


75J07 


67323 


73944 


G8603 


72757 


698C2 


71549 


41 




20 


64723 


76229 


66044 


75088 


67344 


73924 


68(^24 


72737 


69883 


71.529 


40 




21 


64746 


76210 


66066 


75069 


67366 


739;)4 


68645 


72717 


69904 


71508 


39 




22 


647G8 


76192 


66088 


75050 


G7387 


73885 


68666 


72697 


69925 


71488 


38 




23 


64790 


76173 


66109 


75030 


G7409 


73865 


68688 


72677 


69946 


71468 


37 




24 


64812 


76154 


66131 


75011 


67430 


73846 


68709 


72657 


69966 


71447 


36 




25 


64834 


76 J 35 


66153 


74992 


67452 


73826 


68730 


72637 


69987 


71427 


35 




26 


64856 


76116 


66175 


74973 


67473 


73806 


68751 


72617 


70008 


71407 


34 




27 


64878 


76097 


66197 


74953 


67495 


73787 


68772 


72597 


70029 


71386 


33 




28 


64901 


76078 


66218 


74934 


67516 


73767 


68793 


72577 


70049 


71366 


32 




29 


64923 


76059 


66240 


74915 


67538 


73747 


68814 


72557 


70070 


71345 


31 




30 


64945 


76041 


66262 


74893 


67559 


73728 


68835 


72537 


70091 


71325 


30 




31 


64967 


76022 


66284 


74876 


67.580 


73708 


68857 


72517 


70112 


71305 


29 




32 


64989 


76003 


66306 


74857 


67G02 


7::683 


G8S78 


72497 


70132 


71284 


28 




33 


65011 


75984 


66327 


74838 


67623 


736G9 


68899 


72477 


70153 


71264 


27 




34 


65033 


759a5 


66349 


74818 


67645 


73649 


68920 


72457 


70174 


71243 


26 




35 


65055 


75946 


66371 


74799 


67666 


73(:'29 


68941 


72437 


70195 


71223 


25 




36 


iv5077 


75927 


66393 


74780 


67688 


73610 


689G2 


72417 


70215 


71203 


24 




37 


65099 


75D08 


66414 


74760 


67709 


73590 


68983 


72397 


7023(5 


71182 


23 




38 


65122 


75889 


66436 


74741 


67730 


73570 


69004 


72377 


70257 


71162 


22 




39 


65144 


75870 


66458 


74722 


677.52 


73551 


69025 


72357 


70277 


71141 


21 




40 


65166 


75851 


66480 


74703 


67773 


73531 


69046 


72337 


70298 


71121 


20 




41 


65188 


75832 


66501 


74683 


67795 


7351 J 


69967 


72317 


70319: 


71100 


19 




42 


65210 


75813 


66523 


74664 


67816 


73491 


69088 


72297 


70339 71080 


18 




43 


65332 


75794 


66545 


74644 


67837 


73472 


69109 


72277 


70.3(;0 ; 71059 


17 




44 


G5354 


75775 


66566 


74625 


67859 


73452 


69130 


72257 


70381 


71039 


16 




45 


65276 


75756 


66588 


74606 


67880 


73432 


69151 


72236 


70401 


71019 


15 




46 


652D8 


75738 


66610 


74586 


67901 


73412 


69172 


72216 


70422 


70998 


14 




47 


65320 


75719 


66632 


74567 


679.23 


73393 


69193 


72196 


70443 


70978 


13 




48 


65342 


75699 


66653 


74548 


67944 


73373 


69214 


72176 


70463 


70957 


12 




49 


65364 


75680 


66675 


74528 


67965 


73353 


69235 


721.56 


70484 


70937 


11 




50 


65386 


75661 


66697 


74509 


67987 


73333 


69256 


72136 


70.505 


70916 


10 




51 


65408 


75642 


66718 


74489 


68008 


73314 


(59277 


72116 


70525 


7()8i)6 


9 




52 


65430 


75623 


66740 


74470 


68029 


73294 


69298 


72095 


70546 


70875 


8 




53 


65452 


75604 


66762 


74451 


68051 


73274 


69319 


72075 


70567 


70855 


7 




54 


65474 


75585 


66783 


7443] 


68072 


73254 


69340 


72055 


70.587 


70834 


6 




55 


65496 


75566 


66805 


74412 


(58093 


73234 


69361 


72035 


70(>08 


7i!813 


5 




56 


655 J 8 


75547 


66827 


74392 


(>81)5 


73215 


693f^2 


72015 


70(528 


70793 


4 




57 


65540 


75528 


66848 


74373 


6H1.3G 


73195 


69403 


71995 


70(549 


70772 


3 




58 


6">562 


75509 


66870 


74353 


(-.81.57 


73175 


(59-124 i 


71974 


70(570 


70752 


2 




59 


65584 


75490 


66891 


74334 


(;hi79 


731.55 


69445 i 


719.54 


70(590 


70731 


] 






65606 


75471 

'NTs: 


66913 


74314 


(•>8200 


73135 

N.S. 


6946(5 1 


71934 

N.S. 


70711 

N CS. 


70711 

N.S. 







N.CS. 


N.CS. 


N.S. 


N.CS. 


N. CS 1 




49 De^. 1 


48 Deer. 1 


47 Deg. 1 


46 De^r. 1 


45 Deg. 1 



NATURAL TANGENTS. 



99 



M 




De^ 


Trees. 


1 Degree. | 


2 Degrees. | 


3De 
N. Tan. 
05241 


grees. 
N. Cot. 


i 

M 


N. T.in. 


N. Cot. 


N. Tan. 


N. Cot. 


N. Tmd. 
03492 


N. Cot 
23.633.i 


00000 


0000.00 


01746 


.57.2900 


19.0811 


60 ! 


1 


00020 


3437.75 


01775 


56.3503 


03521 


28.399-1 


05270 


18.97.55 


59 


2 


00058 


1718.87 


01804 


55.4415 


03.550 


28.1664 


05299 


18.8711 


58 


3 


00087 


1145.9-2 


018.33 


54.5313 


03579 


27.9372 


0532H 


18.7678 


57 


4 


00116 


859. 43u 


01362 


53.7086 


03609 


27.711-; 


05357 


18.6656 


56 


5 


00145 


687.549 


01891 


52.8321 


03638 


27.4899 


05387 


18.5645 


55 


6 


00175 


572.957 


01920 


52.0307 


03567 


27.2715 


05416 


18-4645 


54 


7 


00234 


491. lOG 


01949 


51.3032 


03696 


27.0566 


05445 


18.3655 


53 


8 


0J233 


429.718 


01978 


50.5485 


03725 


26.8450 


0.5474 


18.2677 


52 


9 


00262 


381.971 


02007 


40.8157 


03754 


26.6367 


0.5503 


18.1708 


51 


10 


00201 


343.774 


02036 


49.1039 


03783 


26.4316 


05533 


18.0750 


50 


11 


00320 


312.521 


02036 


48.4121 


03812 


26.2293 


05562 


17.9801 


49 


12 


00349 


286.478 


0-2095 


47.7305 


03842 


26.0307 


05.591 


17.8863 


48 


13 


00378 


264. 44 J 


02124 


47.08.53 


03871 


25.8348 


05620 


17.7934 


47 


14 


00407 


245.552 


02153 


46.448.) 


03900 


25.6418 


05649 


17.7015 


46 


15 


00436 


229.382 


02182 


45.8294 


03929 


25.4517 


05678 


17.6106 


45 


16 


00465 


214.858 


02211 


45.2261 


03958 


25.2644 


05708 


17.5205 


44 


17 


00495 


202.219 


02240 


.« 4.6336 


03987 


25.0793 


05737 


17.4314 


43 


18 


00524 


190.984 


02-269 


44. 066 L 


04016 


24.8978 


05766 


17.3432 


42 


19 


00553 


180.932 


0-2293 


43.5081 


04046 


24.7185 


05795 


17.25.58 


41 


20 


00582 


171.885 


02328 


42.9641 


04075 


24.5418 


05324 


17.1693 


40 


21 


00611 


183.700 


02357 


42.4335 


04104 


24.3675 


05354 


17.0337 


39 


22 


00640 


156.259 


02333 


41 9158 


04133 


24.1957 


05833 


16.9990 


38 


23 


00669 


149.465 


02415 


41.4103 


04132 


24.0233 


0.5912 


16.91.50 


37 


24 


00698 


143.237 


02444 


40.9174 


04191 


23.8.593 


0.5941 


16.8319 


33 


25 


00727 


137.507 


02473 


40.43.53 


04220 


23.6945 


05970 


16.7493 


35 


26 


00756 


132.219 


02502 


39.9655 


042.50 


23.5321 


05909 


16.6381 


34 


27 


00785 


1-27.321 


0-2.-)31 


39.5059 


04279 


23.3718 


06029 


16.5374 


33 


28 


00314 


122.774 


02530 


39.0.563 


04308 


23.21.37 


00058 


16.5075 


32 


29 


00844 


118.540 


02589 


33.6177 


04337 


23.0.577 


03037 


16.4283 


31 


30 


00873 


114.539 


02619 


38.1885 


04336 


22.9J37 


03116 


16.3499 


30 


31 


00902 


110.892 


02348 


37.7683 


04395 


22.7513 


06145 


16.2722 


29 


32 


00931 


107.42) 


02677 


37.3.579 


04424 


22.0020 


03175 


16.19.52 


28 


33 


00930 


104.171 


0-2705 


36.9530 


014.54 


22.4.541 


03204 


16.1190 


27 


34 


00J89 


101.107 


02735 


35.5327 


01483 


22.3031 


03233 


16.0435 


23 


35 


01018 


93.2179 


02764 


36.1776 


04512 


22.1640 


06232 


15.9387 


25 


36 


01047 


95.4895 


02793 


35.8006 


04541 


22.0-217 


06291 


15.8945 


24 


37 


01076 


92.9085 


02322 


35.4313 


04570 


21.8313 


03321 


15.8211 


23 


38 


01105 


90.4633 


0-2^51 


35.0395 


01599 


21.74-26 


03350 


15.7433 


22 


39 


01135 


83.143G 


0-2831 


34. 715 i 


04;28 


21.6056 


06379 


15.6732 


21 


40 


01164 


85.9393 


02910 


34.3373 


043.58 


21.4704 


06408 


15.6043 


20 


41 


01193 


83.8435 


02J39 


34.0-273 


04387 


21.3369 


06437 


15.5343 


19 


42 


012-22 


81.8470 


02938 


33.6935 


04713 


21.2049 


06437 


15.4333 


IS 


43 


01251 


79.9434 


029^7 


33.3362 


01745 


21.0747 


03493 


15.3943 


17 


1 44 


0128) 


78.1263 


03)23 


33.0452 


04774 


20.9460 


06525 


15.3254 


16 


1 ^^ 


01309 


76.3900 


03055 


32.7303 


04803 


20.3138 


06.554 


15.2571 


15 


46 


01338 


74.7292 


03034 


32.4213 


04332 


20.6932 


06584 


15.1893 


14 


47 


01367 


73.1390 


03114 


32.1181 


1862 


20.. 5691 


06313 


15.12-22 


13 


48 


01396 


71.6151 


03143 


• 31.8205 


04391 


20.4465 


06342 


15.0557 


12 


49 


01425 


70.1.533 


03172 


31.5284 


04920 


20.32.53 


06371 


14.9393 


11 


50 


01455 


63.7501 


03201 


31.2416 


04949 


20.20.53 


06700 


14.9244 


10 


51 


01484 


67.4019 


03230 


30.9599 


04978 


20.0872 


06730 


14.8.593 


9 


52 


01513 


63.1055 


03239 


33.6833 


05097 


19.9702 


06759 


14.7954 


8 


53 


01542 


64.8580 


03233 


30.41:6 


05037 


19.8546 


03788 


14.7317 


7 


54 


01571 


63.6.567 


03317 


30.1446 


05066 


19.7403 


06817 


14.6685 


6 


55 


01600 


62.4992 


03346 


29.8823 


05095 


19.6273 


06847 


14.6059 


5 


56 


01629 


61.3829 


03376 


29.6245 


05124 


19.51.56 


03876 


14.. 5438 


4 


57 


01658 


60.3058 


03405 


29.3711 


051.53 


19.4051 


06905 


14.4323 


3 


58 


01687 


59.2659 


03434 


29.1220 


05182 


19.2959 


06934 


14.4212 


2 


59 


0]716 


58.2612 


03463 


28.8771 


0.5212 


19.1379 


03963 


14.3607 


1 


60 

^^ 

1 


01746 


57.2900 


03492 


23.6363 


05241 
N. Cot. 


19.0811 


06993 


14.3007 


! 

M 


N. Cot. 


N. Tan. 


N. Cot. 


N. Tan. 


N.Tan. 


N. Cot. 


N. Tan. 


89 D 


egrees. 


88 D 


egrees. 


87 D( 


3grees. 


86 Degrees. 



100 



NATURAL TANGENTS. 



M 



4 Degrees. 


5 Degrees. 


6 Degrees. 


7 Degrees. 


n 


.\.Tan. 

0(5993 


N. Cot. 


N.Tan 


N. Cot. 
1174301 


N.Tan. 


N. Cot. 


1 N.Tan 

! 12278 


N. Cot. 


M 


14.3007 


087,^1) 


10510 


9.51436 


8.14435 60 1 


1 


07022 


14.2411 


08778 


11.3919 


10.540 


9.48781 


1 12308 


8.12481 59 1 


2 


07051 


14.1821 


C-3807 


11.3540 


10569 


9.46141 


i 12338 


8.1053C 


58 


3 


07080 


14.1235 


08837 


11.3163 


10599 


9.43515 


1 12367 


8.0860( 


57 


4 


07110 


14.0655 


088CG 


11.2789 


10628 


9.40904 


12397 


8.06674 


56 


5 


07139 


14.0079 


08895 


11.2417 


10657 


9.38307 


12426 


8.0475e 


55 


6 


071G8 


13.9.507 


08925 


11.2048 


10687 


9.35724 


12456 


8.0284c 


54 


7 


07197 


13.8940 


08954 


11.1681 


10716 


9.3.3154 


12485 


8.00948 


53 


8 


07227 


13.8378 


08983 


11.1316 


10746 


9.30599 


12515 


7.99058 


52 


9 


07256 


13.7821 


09013 


11.0954 


10775 


9.280.58 


12.544 


7.97176 


51 


10 


07285 


13.7267 


09042 


11.0594 


10805 


9.2.5530 


12574 


7.95302 


50 


11 


07314 


13.6719 


09071 


11.0237 


10834 


9.23016 


12603 


7.93438 


49 


]2 


07344 


13.6174 


09101 


10.9881 


10863 


9.20566 


12633 


7.91582 


48 


13 


07373 


13.5634 


09130 


10.9.528 


10893 


9.18028 


12662 


7.89734 


47 


14 


07402 


13.5C98 


09159 


10.9178 


10922 


9.1.55.54 


12692 


7.87895 


46 


15 


07431 


13.4566 


09189 


10.8829 


10952 


9.13093 


12722 


7.86064 


45 


16 


07461 


13.4039 


09218 


10.8483 


10981 


9.10646 


12751 


7.84242 


44 


17 


07490 


13.3515 


09247 


10.8139 


11011 


9.08211 


12781 


7.82428 


43 


18 


07519 


13.2U9(- 


09277 


10.7797 


11040 


9.05789 


12810 


7.80622 


42 


19 


07548 


13.2480 


G930G 


10.7457 


11070 


9.03379 


12840 


7.78825 


41 


20 


07578 


13.1969 


09335 


10.7119 


11099 


9.00983 


12869 


7.77035 


40 


21 


07607 


13.1461 


«]9365 


10.6783 


11128 


8.98598 


12899 


7.75254 


39 


22 


07636 


13.09.38 


G9394 


10.6450 


11158 


8.96227 


12929 


7.73480 


38 


23 


07665 


13.0458 


09423 


10.6118 


11187 


8.93867 


12958 


7.71715 


37 


24 


07(595 


12.9962 


09153 


10.5789 


11217 


8.91520 


12988 


7.699.57 


36 


25 


07724 


j2.94(:9 


(-9482 


10.. 5462 


11246 


8.89185 


13017 


7.68208 


35 


20 


07753 


12.8981 


09511 


10.513r 


11276 


8.86862 


13047 


7.66466 


34 


27 


07782 


12.8496 


(9.541 


10.4813 


11305 


8.84551 


13076 


7.64732 


33 


28 


07812 


12.8014 


69570 


10.4491 


11.335 


8.82252 


13106 


7.63005 


32 


29 


07841 


12.7536 


096!. '0 


10.4172 


11364 


8.79964 


13136 


7.61287 


31 


30 


07870 


12.7062 


09629 


10.3854 


11394 


8.77G89 


13165 


7.59575 


30 


31 


07899 


12.6591 


090.58 


10.3538 


11423 


8.75425 


13195 


7.57872 


29 


32 


07929 


12.6124 


C9688 


10.3224 


114.52 


8.73172 


13224 


7.56176 


28 


33 


07958 


12.. 5660 


09717 


10.2913 


11482 


8.70931 


13254 


7.. 54487 


27 


34 


07987 


12.5199 


09746 


10.2602 


11511 


8.68701 


13284 


7.52806 


26 


35 


OSw-n 


12.4742 


09776 


10.2294 


11541 


8.66482 


13313 


7.51132 


25 


35 


08046 


12.4288 


09805 


10.1988 


II57O 


8.(54275 


13343 


7.49465 


24 


37 


08375 


12.38;>8 


09834 


10.1C83 


11 600 


8.62078 


13372 


7.47806 


23 


38 


08104- 


12.3:591] 


C98^:4 


10.1.?81 


11629 


8.. 59893 


13402 


7.46154 


22 


.''lO 


08134 


12.2946 


(.9893 


10.i;;80 


11659 


8.57718 


13432 


7.44509 


21 


40 


08163 


12.2505 


099-23 


10.0780 


11688 


8.55555 


13461 


7.42871 


20 


41 


08192 


12.2;)67 


09952 


10.0483 


11718 


8.53402 


1.3491 


7.41240 


19 


42 


08221 


12.1632 


09981 


10.0187 


11747 


8.512.59 


13521 


7.39616 


18 


43 


08251 


12.1201 


10011 


9.98930 


11777 


8.49128 


13550 


7.. 37999 


17 


44 


08280 


12.0772 


10040 


9.96007 


11806 


8.47007 


13580 


7.36389 


16 


45 


08309 


12.0346 


10069 


9.931U1 


11836 


8.44896 


13609 


7.34786 


15 


4G 


08339 


11.9923 


10099 


9.90211 


11865 


8.42795 


13639 


7.33190 


14 


47 


083r8 


ii.9.5;;4 


10128 


9.87338 


11895 


8.40705 


13669 


7.31600 


13 


48 


08397 


11.9087 


10158 


9.84482 


11924 


8.38(i25 


13698 


7.30018 


12 


4d 


08427 


11.8673 


10187 


9.81641 


11954 


8.36555 


13728 


7.28442 


11 


50 


08456 


11.8262 


1021G 


9.78817 


11983 


8.34496 


137.58 


7.26873 


10 


51 


08485 


11.7853 


10246 


9.76009 


12013 


8.. 32446 


13787 


7.25310 


9 


52 


08514 


11.7448 


10275 


9.73217 


12042 


8.30406 


13817 


7.23754 


8 


53 


08.544 


11.7045 


10:^5 


9.70441 


12072 


8.28376 


13846) 


7.22204 


7 


54 


08573 


11.6645 


10334 


9.67C>8:) 


12101 


8.26355 


13876 


7-20661 





55 


08602 


11.6248 


1036n3 


9.6)4935 


12131 


8.24345 


139;)G 


7.19125 


5 


5() 


08632 


11.. 5853 


10393 


9.62205 


121ii0 


8.22344 


i:;935 


7.17594 


4 


57 


08661 


11.5461 


10422 


9.59490 


12190 


8.203.52 


13965 


7.16071 


3 


58 


08690 


11. 507*2 


104.52 


9.. 56791 


12219 


8.18370 


13995 


7.14553 


2 


59 


08720 


11. 46)85 


10481 


9.54106 


12249 


8.16398 


14024 


7.13042 


i 


GO 
W 


08749 
N Cot. 


11.4301 


10510 


9.51436 
N. Tan. 


12278 


8.14435 
N. Tan. 


14054 
N. Cot. 


7.11537 
N. Tan. 



M 


N. Tan. 


N. Cot. 


N. Cot. 


85 De 


grecs. 


84 De 


grees. 


8315^ 


grees. 


82 De 


grees. 



NATURAL TANGENTS. 



101 



M 
~0 


8 Degrees. | 


9 Degrees. 


10 D 


agrees. 


11 D 


3grees. 


M 

60 


N.Tan. 


N. Cot. 


N. Tai. 

jr.838 


xN". Cot. 
6.31375 


N. Tan. 


N. Cut. 
5. 67 J 2- 


N. Tan 


N. Cot. 


14054 


7.11.537 


171)33 


19438 


5.144,55 


1 


14084 


7.100:}8 


158(38 


6.30189 


17663 


5.66165 


194t)8 


5.13653 


59 


2 


14113 


7.08541-i 


1.5898 


6.29007 


17C93 


5.6.5-205 


19498 


5.12862 


58 


3 


14143 


7.070.59 


1.59-28 


6.278-29 


17723 


5.64243 


19529 


5.12069 


57 


4 


14173 


7.05579 


15958 


6.26655 


17753 


5.63295 


19.5.59 


5.11279 


56 


5 


14202 


7.04105 


15988 


6.2.5486 


17783 


5.62344 


19.589 


5.10490 


55 


6 


14232 


7.02637 


16017 


6.21321 


17813 


5.61397 


19)19 


5.09704 


54 


7 


1421)2 


7.0117-1 


16)47 


6.23160 


17843 


5.604.52 


19649 


5.08921 


53 


8 


14291 


6-99718 


16077 


6.22J0:^ 


17373 


5.59511 


19680 


5.08139 


52 


9 


14321 


6.9S2')8 


16107 


6.20851 


179)3 


5.. 53573 


19710 


5.07360 


51 


10 


14351 


6.96823 


16137 


6.19703 


17933 


5.5:638 


19740 


5.06.584 


50 


11 


14381 


6.9.538.5 


16167 


6.18559 


17963 


5.56706 


19770 


5.05309 


49 


12 


14410 


6.93952 


16196 


6.17419 


17993 


5-5.5777 


19801 


5.05037 


48 


13 


14440 


6.92525 


16226 


6.16283 


18923 


5-54351 


19331 


5.04-267 


47 


14 


14470 


6.91104 


16253 


6.15151 


18053 


5.53927 


19861 


5.03499 


46 


15 


14499 


6-89688 


16-283 


6.14023 


18083 


5-53007 


19891 


5.02734 


45 


16 


14.529 


6-88278 


16316 


6.12899 


-18113 


5.. 52090 


19921 


5.01971 


44 


17 


14559 


6.83874 


16346 


6.11779 


18143 


5.51176 


19952 


5.01210 


43 


18 


14538 


6-85475 


16376 


6.10664 


18173 


5.50-264 


19.J82 


5.00451 


42 


19 


14618 


6.8498-2 


16405 


6.09.55: 


18-203 


5.493.56 


20012 


4.99695 


41 


20 


14048 


6.82(;94 


16435 


6.08444 


18233 


5.48451 


2u042 


4.98940 


40 


21 


14678 


6-81312 


16465 


6.07340 


18263 


5.47543 


20J73 


4.93183 


39 


22 


14707 


6.79936 


16195 


6.03240 


18293 


5.46(543 


20103 


4.97438 


38 


23 


14737 


6.78564 


16525 


6.05143 


18323 


5.4.5751 


20133 


4.96690 


37 


24 


14767 


6.77199 


165.55 


6.0405 i 


18353 


5.44857 


20164 


4.95945 


36 


25 


14793 


6.7.5838 


16585 


0.02962 


18383 


5.439:)6 


20194 


4.95-201 


35 


26 


14826 


6.74483 


16615 


6.0lo7o 


18414 


5.43077 


202-24 


4.94460 


34 


27 


148.55 


6.73133 


16645 


6.00797 


18144 


5.42192 


29254 


4.93721 


33 


28 


14886 


6.7178J 


16674 


5.99720 


18 m 


5.41309 


29-285 


4.9-2984 


32 


29 


14915 


6.704.50 


16704 


5.93646 


185;)4 


5.404-29 


20315 


4.92-249 


31 


30 


14945 


6.69116 


16731 


5.97576 


18534 


5.39552 


20345 


4.91516 


30 


31 


14975 


6.67787 


16764 


5.96510 


18564 


5.33677 


20376 


4.90785 


23 


32 


151)05 


6.66463 


16794 


5.9.5448 


18594 


5.37395 


2)496 


4. 900.50 


23 


33 


15034 


6.65144 


16824 


5.91390 


18624 


5.3393-^ 


2)436 


4.89330 


27 


34 


15J64 


6. 6.383 J 


16854 


5.93335 


18654 


5.36070 


20466 


4.88005 


26 


35 


1.5091 


6.62523 


16884 


5.92-233 


18634 


5.3.5-206 


20497 


4.87882 


25 


36 


15124 


6.61219 


16914 


5.91235 


18714 


5.34345 


20527 


4.87162 


24 


37 


15153 


6.59921 


16944 


5.9.J191 


18745 


5.33437 


29557 


4.86444 


23 


38 


15183 


6.58G27 


16974 


5.89151 


18775 


5.32631 


20588 


4.85727 


22 


39 


15213 


6.57339 


17004 


5.88114 


18895 


5.31778 


20618 


4.85013 


21 


40 


15243 


6.56055 


17033 


5.87030 


18335 


5.30928 


20648 


4.84300 


20 


41 


15272 


6.54777 


17063 


5.86051 


18865 


5.30080 


20679 


4.83590 


19 


42 


15302 


6.53503 


17093 


5.85024 


18895 


5.29-235 


20709 


4.82882 


18 


43 


153.32 


6.52234 


17123 


5.84001 


18925 


5.23393 


29739 


4.82175 


17 


44 


15362 


6.50970 


17153 


5.8-2982 


1-955 


5.27.553 


20770 


4. 8147 i 


16 


45 


15391 


6.49710 


17183 


5.8193G 


18986 


5.26715 


20800 


4.89769 


15 


46 


15421 


6.48456 


17213 


5.80953 


19016 


5.25880 


20830 


4.80068 


14 


47 


1.5451 


6.4720(3 


17243 


5.79944 


19046 


5.25048 


20861 


4.79370 


13 


48 


15181 


6.45961 


17273 


5.78938 


19076 


5.24218 


20891 


4.78673 


12 


49 


155U 


6.44720 


17303 


■ 5.77936 


19106 


5.23391 


20921 


4.77978 


11 


! 50 


15540 


6.43484 


17333 


5.76937 


19136 


5.22566 


20952 


4.77-235 


10 


51 


1.5570 


6.42253 


17363 


5.75941 


19166 


5.21744 


20932 


4.76595 


9 


52 


15600 


6.41026 


17393 


5.74949 


19197 


5.20925 


21013 


4.75903 


8 


53 


15630 


6.39S34 


17423 


5.73960 


19227 


5.20107 


21043 


4.75-219 


7 


54 


15 )60 


6.38587 


17453 


5.72974 


19-257 


5.19293 


21073 


4.74534 


6 


55 


15389 


6.37374 


17483 


5.71992 


19-287 


5.18480 


21194 


4.73851 


5 


56 


1.5719 


6.36165 


17513 


5.71013 


19317 


5.17671 


211.34 


4.73170 


4 


57 


15749 


6.349 3] 


17543 


5.70037 


19347 


5.16863 


21164 


4.72490 


3 


58 


15779 


6.33761 


17.573 


5.69064 


19378 


5.160.58 


21195 


4.71813 


2 


59 


15809 


6..325;5-; 


]7(U)3 


5.68094 


194D8 


5.15256 


21-225 


4.71137 


1 


60 
M 


15838 


6.31375 


17633 


5.671-28 
N.Tan. 

agrees. 


19433 


5.14455 


21256 
N. Cot. 


4.70463 



M 


N. Cot. 


N. Tan. 


N. Col. 
80 D( 


N. Cot. 


N. Tan. 


N. Tan. 


81 Degrees. | 


79 Degrees. I 


78 Degrees. 



102 



NATURAL TANGENTS. 



M 
U 


1*2 Degrees. 


13 D 


egrees. 


14 D 


egrees. 


15 D 


egrees. 


i jM 

' GO 


N.Tan. 
21256 


N. Cot. 


N.Thd 


N. Cot. 


N.Tan. 
24933 


N.Cot. 


N.Tan. 
"26795 


N. Cot. 
3,73205 


4.70463 


23087 


4.33148 


4.01078 


1 


2 J 23:') 


4.69791 


23117 


4.. 32573 


24964 


4.00582 


26826 


3.72771 


59 


2 


21316 


4.6912J 


23148 


4.32001 


24995 


4.00080 


26S57 


3.72338 


58 


3 


21347 


4.G8452 


23179 


4.31430 


25026 


3.99592 


26888 


3.71907 


57 


4 


21377 


4.67786 


23209 


4.3086l< 


25056 


3.99099 


26920 


3.71476 


56 


5 


21408 


4.67121 


23240 


4.30291 


25087 


3.98607 


26951 


3.71046 


55 


6 


21438 


4.664.58 


23271 


4.29724 


25118 


3.98117 


26982 


3.70616 


54 


7 


21469 


4.65797 


23301 


4.291.59 


25149 


3.97627 


27013 


3.70188 


53 


8 


21499 


4.651.38 


23332 


4.28.595 


25180 


3.97139 


27044 


3.69761 


52 


9 


21529 


4.64480 


233()3 


4.28032 


2.5211 


•3.96651 


27076 


3.69335 


51 


10 


21560 


4.63825 


23393 


4.27471 


2.5242 


3.96165 


27107 


3.68909 


50 


]1 


21590 


4.63171 


23424 


4.26911 


2.5273 


3.95680 


27138 


3.68485 


49 


12 


21621 


4.62518 


23455 


4.263.52 


25.304 


3.95196 


27169 


3.68061 


48 


13 


21651 


4.61868 


23484 


4.25795 


25335 


3.94713 


27201 


3.67638 


47 


14 


21682 


4.GJ219 


23516 


4.25239 


25366 


3.94232 


27232 


3.67217 


46 


15 


21712 


4.G0572 


23547 


4.24685 


25397 


3.93751 


272G3 


3.G679G 


45 


IG 


21743 


4.59927 


23578 


4.24132 


25428 


3.93271 


27294 


3.66376 


44 


17 


21773 


4.59283 


23()()8 


4.23580 


25459 


3.92793 


27326 


3-65957 


43 


18 


2J804 


4.5S641 


23639 


4.2.3030 


25490 


3.92316 


27357 , 


3.6.5,538 


42 


]9 


2 J 834 


4.58001 


23670 


4.22481 


25521 


3.91839 


27388 


3.65121 


41 


20 


21864 


4.57363 


23700 


4.21933 


25552 


3.91364 


27419 


3.64705 


40 


21 


21895 


4.56726 


23731 


4.21387 


25.583 


3.90890 


27451 


3.64289 


39 


22 


21925 


4..56C91 


23762 


4.20842 


2,5614 


3.90417 


27482 


3.63874 


38 


22 


21956 


4.554.58 


23793 


4.20298 


25645 


3.89945 


27513 


3.63461 


37 


24 


21986 


4.54826 


23823 


4.19756 


25676 


3.89474 


27545 


3.63048 


36 


25 


22017 


4.5419(; 


23854 


4.19215 


25707 


3.89004 


27576 


3.62636 


35 


26 


22047 


4.53568 


23885 


4.18675 


25738 


3.88,536 


27607 


3.62224 


34 


27 


22078 


4.52941 


23916 


4.18137 


25769 


3.88068 


276)38 


3.61814 


33 1 


2-8 


22108 


4.. 523 16 


23946 


4.17000 


2.5800 


3.87601 


27670 


3.61405 


32 


29 


22139 


4.51693 


23977 


4.17064 


2.5831 


3.87136 


27701 


3.60996 


31 1 


30 


221G9 


4.51071 


24008 


4.1G530 


2.5862 


3.86671 


27732 


3.60588 


30 ■■ 


31 


22200 


4.50451 


24039 


4.15997 


25893 


3.8620« 


27764 


3.60181 


29 i 


32 


22231 


4.49832 


240C9 


4.15465 


25924 


3.85745 


27795 


3.59775 


28 


33 


22261 


4.492J5 


24i00 


4.149.34 


25955 


3.8.5284 


27826 


3.59370 


27 


34 


22292 


4.48i)00 


24131 


4.14405 


25986 


3.84824 


27858 


3.58966 


26 


35 


22322 


4.47986 


24162 


4.i;!877 


26017 


3.843(54 


27889 


3.58562 


25 


36 


22353 


4.47374 


24193 


4.133.50 


26048 


3.83904 


27920 


3.58160 


24 


37 


22383 


4.46764 


24223 


4.12825 


26079 


3. 83449' 


27952 


3.. 577.58 


23 1 


38 


22414 


4.46155 


24254 


4.12301 


26110 


3.829921 


279S3 


3.57357 


22 


39 


22444 


4.45548 


24285 


4.11778 


26141 


3.82537! 


28015 


3.56957 


21 


40 


22475 


4.44942 


24316 


4.112.56 


26172 


3.82083 


28046 


3.56.557 


20 


41 


22505 


4.443.38 


24347 


4.107.36 


26203 


3.816.30 ! 


28077 


3.56159 


19 


42 


22536 


4.43735 


24377 


4.10216 


26235 


3. 8 J 177 


28109 


3.55761 


18 


43 


22567 


4.431.34 


24408 


4.09{;99 


26266 


3.80726 


28140 


3.55364 


17 


44 


22597 


4.42534 


24439 


4.09182 


26297 


3.80276 


28172 


3.54968 


16 


45 


22628 


4.41936 


24470 


4.086G() 


2G328 


3.79827 


28203 


3.54573 


15 


46 


22G58 


4.41340 


24501 


4.08152 


26359 


3.79378 


28234 


3.54179 


14 


47 


22689 


4.40745 


24532 


4.07639 


26390 


3.78931 


28206 


3.53785 


13 


48 


22719 


4.40152 


24562 


4.C7127 


26)421 


3.78485 


28297 


3.53393 


12 


49 


22750 


4.39560 


21593 


4. 006 16 


26452 


3.78040 


28329 


3.53001 


11 


50 


22781 


4.38969 


24624 


4.06107 


2C.483 


3.77595 


28360 


3.52(509 


10 : 


51 


22811 


4.38381 


24(555 


4.0,5599 


26515 


3.77152 


28391 


3.52219 


'J J 


52 


22842 


4.37793 


24686 


4.05092 


26546 


3.76709 


28423 


3.51^•29 


8 


53 


22872 


4.37207 


24717 


4.04586 


265"; 7 


3.76)268 


28454 


3.51441 


7 


54 


22903 


4.36623 


24747 


4.04081 


2())U;8 


3.75828 


23486 


3.51053 


G 


55 


22934 


4.36040 


24778 


4.035781 


2r>639 


3.7.5388 


28517 


3.50666 


5 


56 


22964 


4.3.54.59 


24869 


4.03075! 


2)i670 


3.74950 


28.549 


3.5027o» 


^ 


57 


22995 


4.34879 


24«-10 


4.02.574 


2()70l 


3.74512 


28580 


3.49894 


3 


58 


23;j26 


4.34.300 


24-i71 


4.02074 


2!->733 


3.74075 


28612 


3.49509 


2 


59 


23056 


4.33723 


24902 


4.01.576 


2(5764 


3.73(544 


28(543 


3.49125 


1 


GO 
M 


23087 
N. Cot. 


4.33148 
N.Tan. 


21933 


4.01078 


26795 

N.Cot.i 


3.73205 


286)75 


3.48741 
N. Taa. 


M 


N. Cot. 

~7irrk 


N. Tan. 
grees. 


N. Tan. 


N Cot. 


77 Dr 


(Trees. 


75"De 


rrrees. 


74 Dcgrect?. 



NATURAL TANGENTS. 



103 



M 

~0^ 


16 Degrees. 


! 17 Dc 

N.Tai. 
30 73 


3grees. 
N. Cot. 


18 Degrees, i 


19 Degrees. 


M 

60 


N.Tan. 


N. Cot. 


N Tin. 


i\. Cut. 
3.07768 


\. Tan. 


N. Cot. 


28675 


3. 4874 J 


3.27U85 


324J2 


34433 


2.91)421 


1 


28706 


3.48359 


3;)n05 


3.26745 


3J524 


3.07464 


34465 


2.90147 


59 


2 


28738 


3.47977 


39637 


3.26406 


32556 


3.07160 


34498 


2.«9873 


53 


3 


2876J 


3.4759(5 


30j69 


3.2)067 


32588 


3.06857 


315.30 


2.89600 


57 


4 


28800 


3.472J6 


30700 


3.2572i) 


32621 


3.06654 


34563 


2.89327 


56 


5 


28832 


3.46837 


30732 


3.25392 


32653 


3.062.52 


34596 


2.H9055 


55 


6 


28834 


3.46458 


30764 


3.25055 


32685 


3.05950 


34628 


2.«8783 


54 I 


7 


28895 


3.46080 


30796 


3.24719 


32717 


3.05649 


346G1 


2.88.511 


53 1 


8 


28927 


3.45703 


30823 


3.24383 


32749 


3.05349 


341)93 


2.88240 


52 1 


9 


28J58 


3.45327 


30860 


3.24049 


32782 


3.05049 


34726 


2.87970 


51 1 


10 


289:i0 


3.44951 


30891 


3.23714 


32814 


3.04749 


.347.58 


2.87700 


50 


11 


2^021 


3.44576 


30923 


3.23381 


32846 


3.04450 


34791 


2.87430 


49 


12 


29053 


3.44202 


30955 


3.23048 


32878 


3.04152 


34824 


2.87161 


48 1 


13 


29J34 


3.43829 


30987 


3.22715 


32911 


3.03854 


34856 


2.86892 


47 ! 


! u 


29116 


3.43456 


31019 


3.22:J84 


32943 


3.035.56 


34889 


2.86624 


46 1 


1 15 


29147 


3.43084 


31051 


3.22053 


32975 


3.03260 


34922 


2.86356 


45 


Il6 


23179 


3.42713 


31033 


3.21722 


33007 


3.02963 


34954 


2.86089 


44 


1 l'^ 


29210 


3.?2313 


31115 


3.21392 


33040 


3.02667 


34987 


2.85822 


43 


18 


29212 


3.41973 


31147 


3.21063 


33072 


3.02372 


.35019 


2.85.555 


42 


19 


29274 


3.41694 


31178 


3.2)734 


33104 


3.02077 


35052 


2.9.5289 


41 


20 


29305 


3.412J3 


31210 


3.20406 


33136 


3.01783 


35085 


2.85023 


40 


21 


29337 


3.408)9 


31242 


3.29079 


33169 


3.01489 


35117 


2.84758 


39 


22 


29368 


3.4050-2 


31274 


3.19752 


33201 


3.01190 


35150 


2.84494 


38 


23 


29100 


3.40136 


3 J 306 


3.19426 


33233 


3.00903 


35183 


2.84229 


37 


24 


29i32 


3.39771 


31338 


3.19100 


3.3266 


3.00611 


35216 


2.83905 


36 


25 


29163 


3.39408 


31370 


3.18775 


33298 


3.00319 


35248 


2.83702 


35 


26 


29495 


3.39042 


31402 


3.18451 


.33330 


3.00028 


35281 


2.83439 


34 


27 


29523 


3.38679 


31434 


3. 18127 


33363 


2.99738 


3.5314 


2.83176 


33 


28 


29558 


3.38317 


31466 


3.17804 


33395 


2.99447 


3534(5 


2.82914 


32 


29 


29590 


3.37955 


31498 


3.17481 


33427 


2.99158 


35379 


2.82653 


31 


i 30 


29621 


3.37594 


3153J 


3.17159 


33460 


2.98868 


35412 


2.82391 


30 


1 31 


23053 


3.37234 


31562 


3.18838 


33492 


2.98580 


35445 


2.82130 


29 


32 


29685 


3.36875 


31594 


3.16517 


33524 


2.98292 


35477 


2.81870 


28 


33 


29716 


3.36516 


31626 


3.16197 


33557 


2.98004 


35510 


2.81610 


27 


34 


29748 


3. 36 J 58 


31658 


3.15877 


33589 


2.97717 


35543 


2.81350 


2t> 


j 35 


29780 


3.35800 


31690 


3.15.558 


33621 


2.97430 


35576 


2.81091 


25 


36 


29811 


3.35443 


32722 


3.15210 


3^3654 


2.97144 


35608 


2.80333 


24 


|37 


29843 


3.35087 


31754 


3. 14922 


33686 


2.96858 


35641 


2.80574 


23 


38 


29875 


3.34732 


31786 


3.14605 


33718 


2.96573 


35674 


2.80316 


22 


39 


29906 


3.34377 


31818 


3.14288 


33751 


2.96288 


35707 


2.80059 


21 


40 


29938 


3.34023 


31850 


3.13972 


33783 


2.96004 


35740 


2.79802 


20 


1 41 


29970 


3.35670 


31882 


3.13656 


33816 


2.95720 


35772 


2.79545 


19 


|42 


30001 


3.33317 


31914 


3.13341 


33848 


2.9.5437 


35805 


2.79289 


18 


43 


30)33 


3.32965 


31946 


3.13027 


33881 


2.95)55 


35838 


2.79033 


17 


|44 


30965 


3.32614 


31978 


3.12713 


33913 


2.04872 


3.5871 


2.78778 


10 


145 


30097 


3.32264 


32010 


3.12400 


33945 


2.94590 


35904 


2.78523 


15 


1 46 


30128 


3.31914 


32042 


3.12087 


33978 


2.94309 


35937 


2.78269 


14 


47 


30160 


3.31585 


32074 


3.11775 


34010 


2.94028 


359()9 


2.7SC14 


13 


48 


30192 


3.31216 


32106 


3.11464 


34043 


2.93748 


3u0ii2 


2.77761 


12 


49 


30224 


3.30868 


.321.39' 


3.11153 


34075 


2.931GH 


36035 


2.77507 


11 ! 


50 


3-.)255 


3.30521 


32171 


3.10842 


34108 


2.931n9 


36068 


2.77254 


10 1 


51 


30287 


3.30174 


32203 


3.10532 


34140 


2.92910 


36101 


2.77002 


9 i 


52 


30319 


3.29829 


32235 


3.10223 


34173 


2.92632 


36134 


2.76750 


8 1 


53 


33351 


3.29483 


32267 


3.09914 


34205 


2.92354 


36167 


2.76498 


V| 


1 54 


30382 


3.29139 


32299 


3.09606 


342.18 


2.92076 


36199 


2.76247 


6 1 


1 55 


30414 


3.28795 


32.331 


3.09298 


34270 


2.91799 


36-232 


2.75996 


5 


56 


3044!) 


3.284.52 


32363 


3.08991 


34303 


2.91523 


36265 


2.75746 


4 


57 


30478 


3.28109 


32396 


3.08685 


34335 


2.91246 


36298 


2.75496 


3 


58 


30509 


3.27767 


3-2428 


3.08379 


34368 


2.90971 


36331 


2.75246 


2 


59 


30541 


3.27426 


32460 


3.08073 


34400 


2.90696 


3()3n4 


2.74997 


1| 


60 
M 


30573 


3.27085 


32492 


3.07768 


34433 


2.90421 

N.Tan. 


36397 

N. Cot. 


2.74748 


I 

1 


N. Cot. 


iV. Tan. 


N. Cot. 


N. Tan. 


N. Got. 


N. Tan. 


73 Degrees. 


72 D( 


3grees. i 


71 D( 


3grees. 


70 D 


3grees. 



104 



NATURAL TANGENTS, 



M 

U 


20 Degrees. | 


21 Degrees. 


22 Degrees. 


23 Degrees. 


M 

60 


iN.Tan. 


N. Cot. 


N.Tan 


N. Cot. 
^760509 


N.Tan. 


N.Cot. 
2.47509 


N.Tan. 


N. Cot. 
2.35585 


36397 


2.74748 


38386 


40403 


42447 


1 


36430 


2.74499 


38420 


2.60283 


40436 


2.47302 


42482 


2.35395 


59 


2 


36463 


2.74251 


38453 


2.60057 


4:)470 


2.47095 


42516 


2.35205 


58 


3 


36496 


2.74004 


38487 


2.59831 


40504 


2.46888 


42.551 


2.35015 


57 


4 


36529 


2.73756 


38520 


2.59606 


40538 


2.46682 


42585 


2.34825 


56 


5 


36562 


2.73509 


38553 


2.59381 


40572 


2.46476 


42619 


2.34636 


55 


6 


36595 


2.73263 


38587 


2.59156 


40606 


2.46270 


42654 


2.34447 


54 


7 


36628 


2.73017 


38620 


2.58932 


40640 


2.46065 


42688 


2.34258 


53 


8 


36661 


2.72771 


38654 


2.58708 


40674 


2.45860 


42722 


2.34069 


52 


9 


36694 


2.72526 


38687 


2.58484 


40707 


2.45655 


42757 


2.33881 


51 


10 


36727 


2. 7228 J 


38721 


2.58261 


40741 


2.45451 


42791 


2.33693 


50 


]] 


3676Q 


2.72036 


.38754 


2.58038 


40775 


2.45246 


42826 


2.33505 


49 


12 


36793 


2.71792 


38787 


2.. 57815 


40809 


2.45043 


42860 


2.33317 


48 


]3 


36826 


2.71548 


38821 


2.57593 


40843 


2.44839, 


42894 


2.33130 


47 


14 


36859 


2.71305 


38354 


2.57.371 


40877 


2.44636 


42929 


2.32943 


46 


15 


36892 


2.71062 


38888 


2.5715U 


40911 


2.44433 


42903 


2.32756 


45 


16 


36925 


2.70819 


38921 


2.56928 


40945 


2.44230 


42998 


2.32570 


44 


17 


36958 


2.70577 


38955 


2.56707 


4G979 


2.44027 


43032 


2.32383 


43 


18 


36991 


2.70335 


38988 


2.56487 


41013 


2.43825 


43067 


2.32197 


42 


19 


37024 


2.70094 


39022 


2.. 56266 


41047 


2.43623 


43101 


2.32012 


41 


20 


37057 


2.69853 


39055 


2.56046 


41081 


2.43422 


43136 


2.31826 


40 


21 


37090 


2.69612 


39089 


2.55827 


41115 


2.43220 


43170 


2.31641 


39 


22 


37123 


2.09371 


39122 


2.5.5608 


41149 


2.43019 


43205 


2.31456 


38 


23 


37157 


2.69131 


39156 


2.55389 


41183 


2.42819 


43239 


2.31271 


37 


24 


37190 


2.68892 


39190 


2.55170 


41217 


2.42618 


43274 


2.31086 


36 


25 


37223 


2.G86.53 


39223 


2.54952 


41251 


2.42418 


43308 


2.30902 


35 


26 


37256 


2.68414 


39257 


2.54734 


41285 


2.42218 


43343 


2.30718 


34 


27 


37289 


2.68175 


39290 


2.54516 


41319 


2.42019 


43378 


2.30534 


33 


28 


37322 


2.67937 


39324 


2.54299 


41353 


2.41819 


43412 


2.30351 


32 


29 


37355 


2.67700 


39357 


2.54082 


41387 


2.41620 


43447 


2.30167 


31 


30 


37388 


2.67462 


39391 


2.53865 


41421 


2.41421 


43481 


2.29984 


30 


31 


37422 


2.67225 


39425 


2.53648 


41455 


2.41223 


43516 


2.29801 


29 


32 


37455 


2.669S9 


39458 


2.53432 


41490 


2.41025 


43550 


2.29619 


28 


33 


37488 


2.66752 


39492 


2.53217 


41524 


2.40827 


43585 


2.29437 


27 


34 


37521 


2.66516 


39.526 


2.53001 


41558 


2.40629 


43620 


2.292.54 


26 


35 


37554 


2.66281 


39559 


2.52786 


41592 


2.40432 


43654 


2.29073 


25 


36 


37588 


2.66046 


39593 


2.52,571 


41626 


2.40235 


43689 


2.28891 


24 


37 


37621 


2.65811 


39626 


2.523.57 


41660 


2.40038 


43724 


2.28710 


23 


38 


37654 


2.G5576 


39660 


2.52142 


41694 


2.39841 


43758 


2.28528 


22 


39 


37687 


2.65342 


39694 


2.51929 


41728 


2.39645 


43793 


2.28348 


21 


40 


37720 


2.65J09 


39727 


2.51715 


41763 


2.39449 


43828 


2.28167 


20 


41 


37754 


2.64875 


39761 


2.51502 


41797 


2.39253 


43862 


2.27987 


19 


42 


37787 


2.64642 


39795 


2.51289 


41831 


2.39058 


43897 


2-27806 


18 


43 


37820 


2.64410 


39829 


2.51076 


41865 


2.. 38862 


43932 


2.27626 


17 


44 


37853 


2.64177 


39862 


2.50864 


41899 


2.38668 


43966 


2.27447 


16 


45 


37887 


2.63945 


39896 


2.50652 


41933 


2.38473 


44001 


2.27267 


15 


46 


37920 


2.63714 


39930 


2.50440 


41968 


2.38279 


44036 


2.27088 


14 


47 


37953 


2.63483 


39963 


2.50229 


42002 


2.38084 


44071 


2.26909 


13 


48 


37986 


2.63252 


39997 


2.50018 


42036 


2.37891 


44105 


2.26730 


12 


49 


38020 


2.63021 


40031 


2.49807 


42070 


2.37697 


44140 


2.26552 


11 


50 


38053 


2.62791 


40065 


2.49597 


42105 


2.37504 


44175 


2.26374 


10 


51 


38086 


2.62561 


40098 


2.49386 


42139 


2.37311 


44210 


2.26196 


9 


52 


38120 


2.62332 


40132 


2.49177 


42173 


2.37118 


44244 


2.26018 


8 


53 


38153 


2.62103 


401(56 


2.48967 


42207 


2.36925 


44279 


2.25840 


7 


54 


38186 


2.61874 


40200 


2.48758 


42242 


2.36733 


44314 


2.25663 


6 


55 


38220 


2.61646 


40234 


2.48549 


42276 


2.36541 


44349 


2.25486 


5 


56 


38253 


2.61418 


40267 


2.48340 


42310 


2.36349 


44384 


2.25309 


4 


57 


38286 


2.61190 


40:!01 


2.48132 


42345 


2.36158 


44418 


2.25132 


3 


58 


38320 


2.60963 


40335 


2.47924 


42379 


2.35967 


44453 


2.24956 


2 


59 


38353 


2.60736 


40369 


2.47716 


42413 


2.35776 


44488 


2.24780 


1 


60 
M 


38386 


2.60509 


40403 
N. Cot. 


2.47509 


42447 


2.35585 


44523 
N. Cot. 


2.24604 



M 


N Cot. 


N. Tan. 


N. Tan. 


N. Cot. 


N. Tan. 


N. Tan. 


69 D( 


agrees. 


~68 D( 


3grees. 


67 Degrees. 


66 D( 


agrees. 



NATURAL TANGENTS. 



105 



M 

o" 


24 Degrees. | 


25 Degrees. 1 


26 Degrees. | 


27 Degrees. 


M 

60 




N.Tan. 


N. Cot. 


N.Tan. 


N. Cot. 


N Tm. 


i\. Cot. 
2.05030 


N. Tan 


N. Cot. 




44523 


2.24304 


43631 


2.14451 


48773 


50953 


1.96261 




1 


44558 


2.24428 


46666 


2.14288 


48309 


2.04379 


5U939 


1.96120 


59 




2 


44593 


2.24252 


46702 


2.14125 


48845 


2.04728 


51026 


1.95979 


58 




3 


44627 


2.24077 


46737 


2.13963 


48381 


2.04577 


51033 


1.95833 


57 




4 


44662 


2.2390-2 


46772 


2.13801 


48017 


2.04426 


51(199 


1.95698 


56 




5 


44697 


2.23727 


4f'808 


2.13630 


48953 


2.04276 


51136 


1.9.5557 


55 




6 


44732 


2.23553 


46843 


2.13477 


48989 


2.04125 


51173 


1.95417 


54 




7 


44767 


2.23378 


46879 


2.13313 


40026 


2.03975 


51209 


1.95277 


53 




8 


44802 


2.23204 


46914 


2.13151 


49062 


2.03825 


51246 


1.95137 


52 




9 


44837 


2.23030 


46950 


2. 12993 


49098 


2.03875 


51283 


1.94997 


51 




10 


44872 


2.22857 


4ny85 


2.12832 


40134 


2.03526 


51319 


1.94858 


50 




11 


44907 


2.22683 


47021 


2.12671 


49170 


2.03376 


51356 


1.94718 


49 




12 


44942 


2.22510 


47058 


2.12511 


49206 


2.03227 


51393 


1.94579 


48 




13 


44977 


2.22337 


47092 


2.12350 


49242 


2.03078 


51430 


1.94440 


47 




14 


45012 


2.22164 


47128 


2.12190 


49278 


2.02929 


51467 


1.94301 


46 




15 


45047 


2.21992 


47163 


2.12030 


49315 


2.02780 


51503 


1.94162 


45 




16 


45032 


2.21819 


47199 


2.11871 


49351 


2.02631 


51540 


1.94023 


44 




17 


45117 


2.21647 


47234 


2.11711 


49387 


2.02483 


51577 


1.93885 


43 




18 


45152 


2.21475 


47270 


2.11552 


49423 


2.02335 


51614 


1.93746 


42 




19 


45187 


2.21304 


47305 


2.11392 


49459 


2.02187 


51651 


1.93808 


41 




20 


45222 


2.21132 


47341 


2.11233 


49495 


2.02039 


51688 


1.93470 


40 




21 


45257 


2.20931 


47377 


2.11075 


49532 


2.01891 


51724 


1.93332 


39 




22 


45^92 


2.20790 


47412 


2.10916 


49538 


2.01743 


51761 


1.93197 


38 




23 


45327 


2.20619 


47448 


2.10758 


49604 


2.01596 


51798 


1.93057 


37 




24 


45362 


2.20449 


47483 


2.10300 


49640 


2.01449 


51835 


1.92920 


36 




25 


45397 


2.20278 


47519 


2.10441 


49877 


2.01302 


51372 


1.92782 


35 




26 


45432 


2.20108 


47555 


2.10284 


49713 


2.01155 


51909 


1.92645 


34 




27 


45467 


2.19338 


47590 


2.10126 


49749 


2.01008 


51946 


1.92503 


33 




28 


45502 


2.19769 


47626 


2.09969 


49786 


2.00862 


51983 


1.92371 


32 




29 


45537 


2.19599 


47662 


2.09811 


49822 


2.00715 


52020 


1.922,35 


31 




30 


45573 


2.19430 


47698 


2.09654 


49858 


2.00569 


52057 


1.92098 


30 




31 


45608 


2.19231 


4T733 


2.O0493 


49894 


2.00423 


52094 


1.91962 


23 




32 


45643 


2.1D092 


47769 


2.03341 


49931 


2.00277 


52131 


1.91825 


28 




33 


45378 


2.18J23 


47805 


2.00134 


49967 


2.00131 


52168 


1.91690 


27 




34 


45713 


2.18755 


47840 


2.09328 


50004 


1.99986 


52205 


1.91554 


26 




35 


45748 


2.18587 


47876 


2.08372 


50040 


1.99841 


52242 


1.91418 


25 




36 


4.784 


2.13419 


47912 


2.0S716 


50076 


1.99695 


52279 


1.91282 


24 




37 


45319 


2.18251 


47948 


2.08560 


50113 


1.99550 


52316 


1.91143 


23 




38 


45854 


2.18084 


47984 


2.08405 


50149 


1.99406 


52353 


1.91017 


22 




39 


45889 


2.17916 


48019 


2.08250 


50185 


1.99261 


52390 


1.90876 


21 




40 


45924 


2.17749 


48055 


2.03094 


50222 


1.99116 


52427 


1.90741 


20 




41 


45930 


2.17532 


48091 


2.07939 


502.58 


1.98972 


52464 


1.906O7 


19 




42 


45995 


2.17416 


48127 


2.07785 


50235 


1.93828 


52501 


1.90472 


18 




43 


48030 


2.17249 


48163 


2.07630 


50331 


1.98684 


52538 


1.90337 


17 




44 


46065 


2.17083 


48198 


2.07476 


50368 


1.98540 


52575 


1.90203 


16 




45 


46101 


2.16917 


48234 


2.07321 


50404 


1.93396 


52613 


1.90069 


15 




46 


46136 


2.16751 


48270 


2.07167 


50441 


1.98253 


52(550 


1.83935 


14 




47 


46171 


2.16585 


48306 


2.07014 


50477 


1.98110 


52637 


1.89801 


13 




48 


46208 


2.18420 


48342 


2.03860 


50514 


1.97963 


52724 


1.89687 


12 




49 


46242 


2.13255 


48373 


2.06706 


50550 


1.97823 


52761 


1.89533 


11 




50 


46277 


2.16090 


48414 


2.06553 


50587 


1.97680 


52798 


1.89400 


10 




51 


46312 


2.15925 


48450 


2.03400 


50623 


1.97.538 


52336 


1.89266 


9 




52 


46348 


2.15760 


48486 


2.06247 


50660 


1.97395 


52873 


1.89133 


8 




53 


46:^83 


2.15596 


48521 


2.06094 


50696 


1.97253 


52310 


1.89000 


7 




54 


4()418 


2.15432 


48557 


2.05942 


50733 


1.97111 


52947 


1.88867 


6 




55 


46454 


2.15288 


48593 


2.0.5789 


50769 


1.96969 


.52984 


1.88734 


5 




56 


46489 


2.15104 


48829 


2.05637 


50806 


1.96827 


53024 


1.88602 


4 




57 


46525 


2.14940 


48665 


2.05485 


50843 


1.93685 


53059 


1.88469 


3 




58 


46560 


2.14777 


48701 


2.05333 


50879 


1.93.544 


53096 


1.83337 


2 




' 59 


46595 


2.14614 


48737 


2.05182 


50916 


1.96402 


53134 


1.88205 


1 




60 
M 


46631 


2.14451 


48773 


2.0.5030 


50953 


1.93261 
N.Tan. 


53171 


1.88073 



M 




N. Cot. 


N. Tan. 


N. Cot. 


N. Tan. 


N. Cot. 


N.'75oI 


N. Tan. 




65 Degrees. 


64 D 


egrees. 


63 D 


agrees. 


62 D 


egrees. 





106 



NATURAL TANGENTS. 





1 

M 
.0 


28 Degrees. 


29 Degrees. 


SOD 


egrees. li 31 Degrees. 


i 




N.Tan. 


N. Cot. 
1.88073 


N.Tan 


N.Cot. 

1.80405 


N.Tan. 
57735 


N. Cot. 
1.73205 


N.Tan 


N. Cot. 


"l M 




53171 


5.5431 


60086 


1.66428 60 




1 


53208 


1.87941 


5.5469 


1.80281 


57774 


1.73089 


60126 


1.66318 59 




2 


53246 


1.87809 


55507 


1.80158 


57813 


1.72973 


60165 


1. 66209 i 58 




3 


53283 


1.87677 


5.5545 


1.80034 


57851 


1.72857 


60205 


1.6G09a 


57 




4 


53320 


1.87.546 


55583 


1.79911 


57890 


1.72741 


60245 


1.6.5990 


56 




5 


53358 


1.87415 


55621 


1.79788 


57929 


1.72625 


60284 


1.65881 


55 




6 


53395 


1.87283 


55059 


1.79665 


5796S 


1.72.509 


60324 


1.65772 


54 




7 


53432 


1.87152 


55697 


1.79542 


58007 


1.72393 


60364 


1.65663 


53 




8 


53470 


1.87021 


55736 


1.79419 


58046 


1.72278 


60403 


1.65554 


52 




9 


53507 


1.8S891 


55774 


1.79296 


58085 


1.72163 


60443 


1.65445 


51 




10 


53545 


1.86760 


55812 


1.79174 


58124 


1.72047 


60483 


1.65337 


50 




1] 


53582 


1.86630 


55850 


1.79051 


58162 


1.71932 


60522 


1.65228 


49 




12 


53620 


1.86499 


55888 


1.78929 


58201 


1.71817 


60562 


1-65120 


48 




13 


53657 


3.86369 


55920 


1.78807 


58240 


1.71702 


60602 


1.65011 


47 




14 


53694 


1.86239 


56964 


1.78685 


.58279 


1.71588 


60642 


1.64903 


46 




15 


53732 


1.86109 


56003 


1.78563 


58318 


1.71473 


60G81 


J. 64795 


45 




16 


53769 


1.85979 


56041 


1.78441 


58357 


1.71358 


60721 


1.64687 


44 




17 


53807 


1.85850 


56079 


1.78319 


58396 


1.71244 


60761 


1.64579 


43 




18 


53844 


1.8.5720 


56117 


1.7819S 


58435 


1.71129 


60801 


1.64471 


42 




19 


53882 


1.85591 


56156 


1.78077 


58474 


1.71015 


60841 


1.64363 


41 




20 


53920 


1.85462 


56194 


1.77955 


58513 


1.70901 


60881 


1.64256 


40 




2] 


53957 


1.85333 


56232 


1.778.34 


58552 


1.70787 


60921 


1.64148 


39 




22 


53995 


1-85204 


56270 


1.77713 


58591 


1.70673 


60900 


1.64041 


38 




23 


54032 


1.85075 


56309 


1.77592 


58631 


1.70560 


61000 


1.63933 


37 




24 


54070 


1.84946 


56347 


1.77471 


58670 


1.70446 


61040 


1.63826 


36 




25 


54107 


1.84818 


56385 


1.77351 


58709 


1.70332 


61080 


1.63719 


35 




26 


54145 


].84G89 


56424 


1.77230 


58748 


1.70219 


61120 


1.63612 


34 




27 


54183 


1.8456 J 


5n462 


1.77110 


58787 


1.70106 


611G0 


1.63505 


33 




28 


54220 


1.8443:^. 


56500 


1.76990 


58826 


1.69992 


61200 


1.63398 


32 




29 


54258 


1.84305 


56539 


1.76869 


58865 


1.69379 


61240 


1.63292 


31 




30 


54296 


1.84177 


56577 


1.76749 


58904 


1.C9766 


61280 


1.63185 


30 




31 


54333 


1.84049 


56616 


1.76629 


58944 


1.69353 


61320 


1.63079 


29 




32 


54371 


1.83922 


56054 


1.76510 


58983 


1.C9.541 


61360 


1.62972 


28 




33 


54409 


1.83794 


56693 


1.76390 


59022 


1.G9428 


61400 


1.62866 


27 




34 


54446 


1.83667 


56731 


1.76271 


59061 


1.6.^315 


61440 


1.62760 


26 




35 


54484 


1.83540 


56769 


1.76151 


59101 


1.69203 


61480 


1.62654 


25 




36 


54522 


1.83413 


56808 


1.76032 


59140 


1.69091 


61520 


1.62548 


24 




37 


54560 


1.83286 


56846 


1.75913 


59179 


1.08979 


61.561 


1.62442 


23 




38 


54597 


1.83159 


56885 


1.75794 


59218 


1.68866 


61601 


1.62.3.36 


22 




39 


546.35 


]. 83033 


56923 


1.75675 


59258 


1.C8754 


61641 


1.62230 


21 




40 


54673 


1.82906 


57962 


1.75556 


59297 


1.68643 


61G81 


1.62125 


20 




41 


54711 


1.82780 


57000 


1.75437 


59338 


1.68531 


61721 


1.62019 


19 




42 


54748 


1.82654 


57039 


1.75319 


59376 


1.G8419 


61761 


1.61914 


18 




43 


54786 


1.82528 


57078 


1.75200 


59415 


1.68308 


61801 


1.61808 


17 




44 


54824 


1.82402 


57116 


1.75082 


59454 


1.68196 


61842 


1.61703 16 II 




45 


54862 


1.82276 


57155 


1.74964 


59494 


1.08085 


61882 


1.61598 


15 




46 


54900 


1.82150 


57193 


1.74846 


59533 


1.67974 


61922 


1.61493 


14 




47 


54938 


1.82025 


57232 


1.74728 


59573 


1.67869 


61962 


1.61388 


13 




48 


54975 


1.81899 


57271 


1.74610 


59:;i2 


1.67752 


62003 


1.61283 


12 




49 


55013 


1.81774 


57309 


1.74492 


59^51 


1.67641 


62043 


1.61179 


11 




50 


55051 


1.81649 


57348 


1.74375 


59691 


1.67530 


62;;83 


1.61074 


10 




51 


55089 


1.81.524 


57386 


1.74257 


59730 


1.67419 


62124 


1.60970 


9 




52 


55127 


1.81.399 


57425 


1.74140 


59770 


1.67309 


62164 


1.60865 


8 




53 


55165 


1.81274 


57464 


1.74022 


59809 


1.67198 


62204 


1.60761 


7 




54 


55203 


1.81150 


57.503 


1.73905 


59849 


1.67088 


62245 


1.60657 


6 




55 


5.5241 


1.81025 


57,541 


1.73788 


59888 


1.66978 


62285 


1.605.53 


5 




56 


5.5279 


1.80901 


57580 


1.73671 


59928 


i.6(-8r>7 


62325 


1.60449 


4 




57 


.55317 


1.80777 


57619 


1.73.5.55 


59967 


1.667.57 


623G6 


1.60345 


3 




58 


55355 


1.806.53 


57(i.57 


1.734.38 


60007 


1.666471 


62-106 


1.60241 


2 




59 


5.5393 


1.80.529 


57()96 


1.73321 


60046 


1.66538 


62446 


1.60137 


1 




60 


55431 


1.80105 
N. Tan 


.57735 
N. Cot. 


1.73205 


00086 


1.66428 


62487 


1.60033 



M 




N Cot. 


N. Tan. 


N. Cot. 


N. Tan. 


N. Cot. 


N. Tan. 




61 De 


.grees. 


60 Degrees. 


59 De 


ffrees. 


58 Degrees. 



NATURAL TANGENTS. 



107 



M 




32 Degrees. 


33 Degrees. 


34 D 


egrees. 


35 D 


egrees. 


M 

60 




nTt^ 


N. Cot. 


N.Tan. 
64941 


N. Cot. 
1.53986 


N.Tan. 
67451 


N. Cot. 
1.48256 


N.Tan. 
~7002r 


N. Cot. 
1.42815 




62487 


1.6UU3.1 




1 


62527 


1.59930 


64982 


1.53888 


67493 


1.48163 


700iH 


1.42726 


59 




o 


62568 


1.5982r. 


65023 


1.. 53791 


67536 


1.48070 


70107 


1.42638 


58 




3 


6-2608 


1.59723 


65035 


1.53693 


67578 


1.47977 


70151 


1.42.550 


57 




4 


62649 


1.59620 


65106 


1.53595 


67620 


1.47885 


70194 


1.42462 


56 




5 


62689 


1.53517 


65148 


1.53497 


676G3 


1.47792 


702.38 


1.42374 


55 




G 


62730 


1.59414 


65189 


1.. 53400 


67705 


1.47699 


70231 


1.42236 


54 




7 


62770 


1.59311 


65231 


1.53302 


67748 


1.47607 


70325 


1.42193 


53 




8 


62811 


1.59203 


65272 


1.53205 


C7790 


1.47514 


70368 


1.42110 


52 




9 


62852 


1.59105 


65314 


i. 53107 


67832 


1.47422 


70112 


1.42022 


51 




10 


62892 


1.59002 


65355 


1.. 530 10 


67875 


1.47330 


70455 


1.41934 


50 




IJ 


62933 


1.58900 


65397 


1.52913 


67917 


1.47238 


70199 


1.41847 


49 




12 


62973 


1.53797 


65438 


1.52816 


67960 


1.47146 


70.542 


1-417.59 


48 




13 


63014 


1.53395 


65480 


1.52719 


68002 


1.470.53 


70586 


1.41672 


47 




14 


63055 


1.53593 


65521 


1.52622 


68045 


1.4G9G2 


70629 


1.41584 


46 




15 


63095 


1.58490 


65563 


1.52525 


68088 


1.46870 


70373 


1.41497 


45 




16 


63135 


1.58388 


6.5801 


1.52429 


68130 


1.46778 


70717 


1.41409 


44 




17 


63177 


1.5828G 


65343 


1.52332 


68173 


1.46G86 


70760 


1.41322 


43 




18 


63217 


1.. 53184 


65G88 


1.. 52235 


68215 


1.46595 


70804 


1.412.35 


42 




19 


63258 


1.53J83 


65729 


1.52139 


68258 


1.48503 


70848 


1.41148 


41 




20 


63299 


1.57931 


65771 


1.52043 


68301 


1.46411 


70891 


1.410G1 


40 




21 


63.140 


1.5787S 


65813 


1.51943 


68343 


1.46320 


70935 


1.40974 


39 




22 


63380 


1.57778 


65854 


1.518.50 


68386 


1.46229 


70979 


1.40887 


38 




23 


63421 


1.. 57676 


65895 


1.51754 


68429 


1.46137 


71023 


1.40800 


37 




24 


634G2 


1.57575 


65938 


1.51G58 


68471 


1.4304G 


71068 


1.40714 


36 




25 


63503 


1.57474 


65930 


1.51.532 


68514 


1.45955 


71110 


1.40627 


35 




2f) 


63544 


1.57372 


66021 


1.51466 


68557 


1.4.5834 


711.54 


1.40540 


34 




27 


63584 


1.57271 


66063 


1.51370 


68600 


1.45773 


71198 


1.40454 


33 




28 


63625 


1.57170 


66105 


1.51275 


6aj42 


1.45382 


71242 


1.40368 


32 




29 


63666 


1.57069 


66147 


1.51179 


68G85 


1.45592 


71285 


1.40281 


31 




30 


63707 


1.5G939 


66189 


1.51084 


68728 


1.45501 


71329 


1.40195 


30 




31 


63748 


1.56863 


66230 


1.50988 


63771 


1.45410 


71373 


1.40109 


29 




32 


63789 


1.53767 


66272 


1.50893 


68314 


1.45320 


71417 


1.40022 


28 




33 


63830 


1.5G667 


66314 


1.50797 


68357 


1.45229 


71461 


1.39936 


27 




34 


63871 


1.56566 


66356 


1.50702 


68900 


1.45139 


71505 


1.39350 


26 




35 


63912 


1.53466 


66398 


1.50607 


()8942 


1.45048 


71549 


1.39764 


25 




36 


63953 


1.56336 


66440 


1.50512 


88935 


1.449.58 


71593 


1.39879 


24 




37 


63994 


1.5;)265 


68482 


1.50417 


69928 


1,44868 


71637 


1.39.593 


23 




38 


64035 


1.56165 


63524 


1.50322 


69071 


1.44778 


71681 


1.39507 


22 




39 


64076 


1.. 53035 


66566 


1.50223 


69114 


1.44688 


71725 


1.39421 


21 




40 


64117 


1.55966 


66G08 


1.50133 


69157 


1.44598 


71769 


1.3933G 


20 




41 


6-1158 


1.55866 


66G.50 


1.50038 


69200 


1.44508 


71813 


1.39250 


19 




42 


64199 


1.55766 


66G92 


1.49944 


69243 


1.44418 


71857 


1.39165 


18 




43 


64240 


1.55366 


6G734 


1.49849 


69286 


1.44329 


71901 


1.39079 


17 




44 


64231 


1.55567 


66776 


1.49755 


69329 


1.44239 


71:446 


1.33994 


16 




45 


64322 


1.55467 


66818 


1.496G1 


G9372 


1.44149 


71990 


1.38909 


15 




40 


64363 


1.55368 


66860 


1.49536 


69416 


1.44000 


72034 


1.38824 


14 




47 


64404 


1.55269 


66902 


1.49472 


694.59 


1.43970 


72078 


1.38738 


13 




48 


64446 


1.55170 


66944 


1.49378 


69502 


1.43881 


72122 


1.38353 


12 




49 


64487 


1.55071 


6693'6 


1.49284 


69545 


1.43792 


7216G 


1.38538 


11 




50 


64528 


1.54972 


67028 


1.49190 


69588 


1.43703 


72211 


1.38484 


10 




51 


64569 


1.54873 


67071 


1.49097 


69631 


1.43614 


722^.55 


1.38399 


9 




52 


64610 


1.54774 


67113 


1.49103 


69G75 


1.43525 


72299 


1.38314 


8 




53 


64652 


1.54675 


67155 


1.48909 


69718 


1.43436 


72344 


1.38229 


7 




54 


64693 


1.54576 


67197 


1.48816 


69761 


1.4.3347 


72388 


1.38145 


6 




55 


64734 


1.54478 


67239 


1.48722 


69804 


1.43258 


72432 


1.38060 


5 




56 


64775 


1.54379 


67282 


1.48629 


69847 


1.43169 


72477 


1.37976 


4 




57 


64817 


1.54281 


67324 


1.48536 


69391 


1.43080 


72521 


1.37891 


3 




58 


64858 


1.54183 


67366 


1.48442 


69934 


1.42992 


72565 


1.37807 


2 




59 


64899 


1.54085 


67409 


1.48.349 


69977 


1.42903 


72310 


1.37722 i 1 




60 
M 


64941 


1.53986 


67451 
N. Cot. 


1.48256 
N. Tan. 


70021 
N. Cot. 


1.42815 


72654 


1.37638 


! 
M 




N Cot. 


N. Tan. 


N. Taa. 


N. Cot. 


N. Tan. 




57 De 


grees. 


56 De< 


^rees. 


55 De 


grees. 


54 De 


jgrees. 





108 



NATURAL TANGENTS. 



JM 



36 Degrees. 


37 D 


egrees. 


38 D 

N. Tan. 


egrees. 
N. Cot. 


39 Degrees. 


M 
60 


N.Tan. 

72654 


N. Cot. 


N.Tan. 


N. Cot. 


N. Tan 
8U978 


N. Cot. 


1.37638 


75355 


1.32704 


78129 


1.27994 


1.23489 


1 


72699 


1.37554 


75401 


1.32624 


78175 


1.27917 


811/27 


1.23416 


59 


2 


72743 


1.37470 


75447 


1.32.544 


78222 


1.27841 


81075 


1.23343 


58 


3 


72788 


1.37386 


75492 


1.32464 


78269 


1.27764 


81123 


1.23270 


57 


4 


72832 


1.37302 


755:i8 


1.32384 


78316 


1.27688 


81171 


1.23196 


56 


5 


72877 


1.37218 


75584 


1.32304 


78363 


1.27611 


81220 


1.23123 


55 


6 


72921 


1.37134 


75629 


1.32224 


78410 


1.27.535 


812G8 


1.230.50 


54 


7 


72966 


1.37050 


75675 


1.32144 


78457 


1.27458 


81316 


1.22977 


53 


8 


73010 


1.36967 


75721 


1.32064 


78504 


1.27382 


8l3f:4 


1.22904 


52 


9 


73055 


1.36883 


75767 


1.31984 


78551 


1.273C6 


81413 


1.22831 


51 


10 


73100 


1.36800 


75812 


1.31904 


78598 


1.27230 


81461 


1.22758 


50 


11 


73144 


1.30716 


75858 


1.31825 


78645 


1.271.53 


81510 


1.22685 


49 


12 


73189 


1.3G633 


75904 


1.31745 


78692 


1.27077 


81558 


1.22612 


48 


13 


73234 


1.36549 


75950 


1.31066 


78739 


1.27001 


81606 


1.22539 


47 


14 


73278 


1.36466 


75996 


1.31586 


78786 


1.26925 


81655 


1.22467 


46 


15 


73323 


1.36383 


76042 


1.31507 


78834 


1.26849 


81703 


1.22394 


45 


16 


73368 


1.36300 


76088 


1.31427 


78881 


1.26774 


81752 


1.22321 


44 


17 


73413 


1.36217 


76134 


1.31348 


78928 


1.26698 


81800 


1.22249 


43 


18 


73457 


1.36133 


76180 


1.31269 


78975 


1.26622 


81849 


1.22176 


42 


19 


73502 


1.36051 


76226 


1.31190 


79022 


1.26546 


81898 


1.22104 


41 


20 


73547 


1.35968 


76272 


1.31110 


79070 


1.26471 


81946) 


1.22031 


40 


21 


73592 


1.35885 


76318 


1.31031 


79117 


1.26395 


81995 


1.21959 


39 


22 


73037 


1.35802 


76364 


1.30952 


79104 


1.26319 


82044 


1.21886 


38 


23 


73681 


1.35719 


76410 


1.30873 


79212 


1.26244 


82092 


1.21814 


37 


24 


73726 


1.35637 


76456 


1.30795 


79259 


1.26169 


82141 


1.21742 


36 


25 


73771 


1.35554 


76502 


1.30710 


79306 


1.26093 


82190 


1 21670 


35 


26 


73816 


1.35472 


76548 


1.301)37 


79354 


1.26018 


82238 


1.21598 


34 


27 


738J6 


1.35389 


76594 


1.30558 


79401 


1.25943 


82287 


1.21526 


33 


28 


73906 


1.35307 


76640 


1.30480 


79449 


1.25867 


82336 


1.21454 


32 


29 


73951 


1.35224 


76686 


1.30401 


79496 


1.25792 


82385 


1.21382 


31 


30 


73996 


1.35142 


76733 


1.30323 


79544 


1.25717 


82434 


1.21310 


30 


31 


74041 


1.35060 


76779 


1.30244 


79591 


1.25642 


82483 


1.21238 


29 


32 


74U86 


1.34978 


76825 


1.30166 


79639 


1.25567 


82531 


1.21106 


28 


33 


74131 


1.34896 


76871 


1.30087 


79686 


1.2549-2 


82580 


1.21094 


27 


34 


74176 


1.34814 


76918 


1.30009 


79734 


1.25417 


82629 


1.21023 


26 


35 


74221 


1.34732 


76964 


1.29931 


79781 


1.25343 


82678 


1.20951 


25 


36 


74267 


1.34650 


77010 


1.298.53 


79829 


1.25268 


82727 


1.20879 


24 


37 


74312 


1.34568 


77057 


1.29775 


79877 


1.25193 


82776 


1.20808 


23 


38 


74357 


1.34487 


77103 


1.29696 


79924 


1.25118 


82825 


1.20736 


22 


39 


74402 


1.34405 


77149 


1.29618 


79972 


1.25044 


82874 


1.20665 


21 


40 


74447 


1.34323 


77196 


1.29541 


80020 


1.24969 


82923 


1.20593 


20 


41 


74492 


1.34242 


77242 


1.29463 


80067 


1.24895 


82972 


1.20522 


19 1 


42 


74538 


1.34160 


77289 


1.29385 


80115 


1.24820 


83022 


1.20451 


18 . 


43 


74583 


1.34079 


77335 


1.29307 


80163 


1.24746 


83071 


1.20379 


17 


44 


74628 


1.33998 


77382 


1.29229 


80211 


1.24672 


83120 


1.20308 


10 


45 


74674 


1.33916 


77428 


1.29152 


80258 


1.24597 


83169 


1.20237 


15 


46 


74719 


1.33835 


77475 


1.29074 


80306 


1.24523 


83218 


1.20166 


14 


47 


74764 


1.33754 


77521 


1.28997 


80354 


1.24449 


83268 


1.20095 


13 


48 


74810 


1.33673 


77568 


1.28919 


80402 


1.243/5 


83317 


1.20024 


12 


49 


74855 


1.33592 


77615 


1.28842 


80450 


1.24301 


83366 


1.19953 


11 


50 


74900 


1.33511 


77661 


1.28764 


80498 


1.24227 


83415 


1.19882 


10 


51 


74946 


1.33430 


77703 


1.28687 


80546 


1.24153 


83465 


1.19811 


9 


52 


74991 


1.33349 


77754 


1.28610 


80594 


1.24079 


83514 


1.19740 


8 


53 


75037 


1.33268 


77801 


1.28533 


80642 


1.24005 


83564 


1.19669 


7 


54 


75082 


1.33187 


77848 


1.284.5(i 


80690 


1.23931 


83613 


1.19.599 


6 


55 


75128 


1.33107 


77895 


1.28379 


80738 


1.23858 


83662 


1.19528 


5 


56 


75174 


1.33026 


77941 


1.28302 


80786 


1.23784 


83712 


1.194.57 


4 


57 


75219 


1.32946 


77988 


1.28225 


80834 


1.23710 


83761 


1.19387 


3 


58 


75264 


1.32865 


77035 


1.28148 


80882 


1.23637 


8381 1 


1.19316 


2 


59 


75310 


1.32785 


77082 


1.28071 


80930 


1.23.563 


83860 


1.19246 


1 


GO 
M 


75355 


1.32704 


77129 


1.27994 


80978 


1.2,3490 
N. Tan. 


83910 


1.19175 



M 


N. Cot. 


N.Tan. 


N. Cot. 


N. Tan. 


N. Cot. 


N. Cot. 

50 D 


N. Tan. 
egrees. 


53 De 


.grees. 


52 D 


3grees. 


51 D 


sgrees. 



NATURAL TANGENTS. 



109 



M 




40 Degrees. 


41 Degrees. 


42 Degrees. 


43 Degrees. 


60~ 




N.Tan. 


N. Cot. 


N.Tan. 


N. Cot. 
1.15037 


N.Tan. 


N. Cot. 


N.Tan. 
"93252" 


JV. Cot. 
1.072^ 




83910 


1.19175 


86929 


90040 


1.11061 




1 


839S0 


1.19105 


86980 


1.14969 


90093 


1.10996 


93306 


1.07174 


59 




o 


84009 


1.19035 


87031 


1.14902 


90146 


1.10931 


93360 


1.07112 


58 




3 


84059 


1.18964 


87J82 


1.14834 


9'J 199 


1.10867 


93415 


1.07049 


57 




4 


84108 


1.18894 


87133 


1.14767 


9u251 


1.10802 


93469 


1.06987 


56 




5 


84158 


1.18324 


87184 


1.14699 


9a3J4 


1.10737 


93524 


1.06925 


55 




6 


84208 


1.18754 


87236 


i.i4;;32 


90357 


1.10672 


93578 


1.06862 


54 




7 


84-258 


1.18684 


87287 


1.14565 


90410 


1.10607 


93033 


1.06300 


53 




8 


84307 


1.18614 


87338 


1.14498 


90463 


1.10543 


93633 


1.06733 


52 




9 


84357 


1.18544 


87389 


1.14130 


905 16 


1.10473 


93742 


1.06676 


51 




10 


84407 


1.18474 


87441 


1.14363 


90569 


1.10414 


93797 


1.06613 


50 




IJ 


84457 


1.18404 


87492 


1.14296 


90621 


1.10349 


93852 


1.06551 


49 




12 


84507 


1.18334 


87543 


1.1422 J 


90674 


1 . 10235 


93906 


1.06489 


48 




13 


84556 


1.18264 


87595 


1.14162 


90727 


1.10-220 


93961 


1.06427 


47 




14 


84606 


1.18191 


87046 


1.14095 


90781 


1.10156 


94016 


1.06365 


46 




15 


84656 


1.18125 


87098 


1.14028 


90834 


1.10091 


94071 


1.06303 


45 




16 


84706 


1.18055 


87749 


1.13961 


90337 


1.10027 


94125 


1.06241 


44 




17 


84756 


1.17986 


87801 


1.13894 


90J40 


1.09J63 


94180 


1.06179 


43 




18 


84806 


1.17916 


87852 


1.13328 


90993 


1.09899 


94235 


1.06117 


42 




19 


84856 


1.17846 


87904 


1.13761 


91046 


1.03334 


94-290 


1.06056 


41 




20 


84906 


1.17777 


87955 


1.13694 


91099 


1.09770 


94345 


1.05994 


40 




21 


84956 


1.17708 


83007 


1.13627 


911.53 


1.09706 


94400 


1.05932 


39 




22 


85006 


1.17638 


88059 


1.13561 


9J2J6 


1.09642 


94455 


1.05870 


38 




23 


85057 


1.17569 


88110 


1.13494 


9 J 2.59 


l.O.t.578 


94510 


1.05809 


37 




24 


85 J 07 


1.1750a 


83162 


1.134-28 


913J3 


1.09514 


94565 


1.0.5747 


36 




25 


85157 


1.17430 


88-214 


1.13361 


91366 


1.0J450 


94620 


1.0.5685 


35 




26 


85207 


1.17361 


88-265 


1.13295 


91419 


1.0.3:386 


94')76 


1.05624 


34 




27 


85257 


1.1729-2 


88317 


1 . 132-28 


91473 


1.0932 J 


94731 


1.0.5562 


33 




28 


85307 


1.17223 


883-69 


1.13162 


91.526 


1.09258 


94786 


1.0.5501 


32 




29 


85358 


1.17154 


88421 


1.13^)96 


91580 


1.09195 


94841 


1.0.5439 


31 




30 


85408 


1.17085 


88473 


1.13029 


91633 


1.09131 


94896 


1.05378 


30 




31 


854.58 


1.17016 


88524 


1.12963 


91687 


1.09067 


94952 


1.05317 


29 




32 


85509 


1.16917 


88576 


1.12807 


91740 


1.09)03 


95;)07 


1.0.5255 


28 




33 


85559 


1.16878 


88628 


1.12831 


91794 


1.08940 


95062 


1.05194 


27 




34 


85609 


1.16809 


88680 


1.12765 


91847 


1.08376 


95113 


1.05i:i3 


26 




35 


85660 


1.16741 


88732 


1.1-2699 


9190] 


1.08813 


95173 


1.0.5072 


25 




36 


85710 


1.16672 


88784 


1.12633 


919.55 


1.08749 


95229 


1.0.5010 


24 




37 


85761 


1.16603 


88:<36 


1.12557 


920U8 


1.08636 


95-284 


1.04949 


23 




38 


85811 


1.16535 


88888 


1.12501 


92062 


1.08622 


95340 


1.01888 


22 1 




39 


85H62 


l.l()46t) 


88940 


1.12435 


92116 


1.08.5.59 


95395 


1.043-27 


21 




-10 


85912 


1.16398 


88992 


1.12369 


92170 


1.084.J6 


95451 


1.04766 


20 




4i 


85963 


1.16329 


89045 


I . 12303 


92223 


1.084.32 


95506 


1.04705 


19 




42 


86014 


1.16261 


89097 


1.12238 


92277 


1.083 '.9 


95562 


1.04644 


18 




43 


86064 


1.16192 


89149 


1.12172 


92331 


1.08:^06 


95618 


1.04.583 


17 




44 


80115 


1.16124 


89201 


1.12106 


92385 


1.03243 


95673 


1.04522 


16 




45 


86166 


1.16056 


89253 


1.12041 


92439 


1.08170 


95729 


1.04461 


15 




46 


86216 


1.15987 


89306 


1.11975 


92493 


1.08116 


95785 


1.04401 


14 




47 


8G2i)7 


1-1.5919 


89353 


1.11909 


9 2.547 


1.080.53 


9.5841 


i.04:mo 


13 




48 


86318 


1.1.5851 


8^410 


1.11844 


92501 


1.07990 


9.5897 


1.04-279 


12 




49 


86368 


1.1.578:? 


89463 


1.11778 


92655 


1.07927 


969.52 


1.04218 


11 




50 


86419 


1.1.5715 


89515 


1.11713 


' 92700 


1. 07864 


96008 


1.041.58 


10 




51 


86470 


1.15647 


89567 


1 . 1 1648 


: 92763 


1.07801 


96064 


1.04097 


9 




52 


86521 


1.1.5.579 


89620 


1.11582 


\ 92817 


1.077:J8 


98120 


1.04036 


8 




53 


88572 


1.15511 


89672 


1. 11.517 


I 92872 


1.07676 


96176 


1.03976 


7 




54 


86623 


1.15443 


89725 


1.114.52 


1 9292() 


1.07613 


96232 


1.03915 


6 




55 


86674 


1.1.5375 


89777 


1.11337 


92930 


1.07.5.50 


96-288 


1.038.55 


5 




56 


8^725 


1.1.5308 


89830 


1.11321 


1 9:5034 


1.07487 


96344 


1.0:3794 


4 




57 


86776 


1.15240 


89833 


1.11256 


1 93088 


1.07425 


96400 


1.0373' 


3 




58 


8r^827 


1.15172 


83935 


1.11191 


I 9:a43 


1.07362 


96457 


1.03674 


2 




59 


86878 


1.1.5104 


89988 


1.11126 


1 93197 


1.07290 


96513 


1.03613 


1 




60 

i 

1 


86929 


] . 1.5037 


90040 
N. Cot. 


1.11061 
N. Tan. 


1 9:^252 


1.07237 


96569 


1.03553 







N Cot 

49 Bi 


N. Tan. 
igrees. | 


N. Cot. 


N. Tan. 


N. Cot. 


N.Tan. 


M 




48 D^ 


ffrees. 


47 Degrees. 1 


46 D 


agrees. 





110 



NATURAL TANGENTS. 



_M 

1 
2 

3 

4 
5 
6 
7 
8 
9 

10 

11 

12 

13 

14 

15 



44 Degrees, 



N/J'an. N.Col. 

UGoGO 
90G-25 
9GG81 
9i3738 
9G794 
9G850 
9G907 
9Gy63 
97020 
97076 
97133 
97189 
97246 
97302 
97359 
97416 



97472 
97529 
9758(i 
97643 
97700 
97756 
97813 
97870 
97927 
97984 
98041 
98098 
98155 

29 I 98213 

30 9S270 

M" N. Cot. 



1.03553 

1.03493 

1.03433 

1.03372 

1.03312 

1.03252 

1.03192 

1.03132 

1.03072 

1.03012 

1.02952 

1.02892 

1.02832 

1.02772 

1.02713 4(3 

1.02653 45 



1.02593 
1.02.533 
1.02474 
1.02414 
1.02355 
1.02295 
1.02236 
1.02176 
]. 02117 
1.02057 
1.01998 
1.01939 
1.01879 
1.01820 
1.01761 



N. Tan. 



45 Degrees. 



M 



31 

32 

33 

34 

35 

36 

37 

3^ 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 



M 



44 Degrees. 



N. Tan. 



98327 
98384 
98441 
98499 
98556 
98613 
98G71 
96728 
98786 
98843 
98901 
98958 
99016 
99073 
99131 
99189 

99247 
99304 
993G2 
99420 
99478 
99536 
99594 
99652 
99710 
99768 
99826 
99884 
99942 
10000 



N.Cot. 



I\\ Cot. 



1.01702 
1.01G42 
1.01583 
1.01524 
1.014G5 
1.01406 
1.01347 
1.012S8 
1.0 1229 I 
1.01170 
1.01112 
1.01053 
1.00994 
1.00935 
1.00876 
1.00818 

1.00759 
1.00701 
1.00642 
1.00583 
1.00525 
1.00467 
1.00408 
1.00350 
1.00291 
1.00233 
1.00175 
l.OOllG 
1.00058 
1.00000 



N. Tan. 



45 Dejirees. 



]Vl_ 

29 
28 
27 
26 
25 
24 
23 
22 
21 
20 
19 
18 
17 
16 
15 
14 

13 

12 
11 
10 
9 
8 
7 
6 
5 
4 
3 
2 
1 




Part II. 
test problems, 



ARITHMETICAL PROBLEMS. 
DENOMINATE NUMBERS. 

1. A printer used 3 reams, 5 quires, 19 sheets of paper for 
printing half-sheet posters. How many did he print, allow- 
ing 1 quire to a ream for waste ? 

2. If a grocer's w^eights are one fourth of an ounce in a 
pound below the legal standard, how much does he gain fraud- 
ulently from the sale of 2 bags of Eio coffee, 116 pounds each, 
true weight, at ISf cents a pound ? 

3. Which is heavier, — a pound of gold, or a pound of lead ; 
an ounce of gold, or an ounce of lead ? 

4. A man starts from Philadelphia, and travels westward. 
When he stops, he ascertains that his watch is 6^ hours slow. 
In what longitude did he stop, and through how many degrees 
did he travel ? 

5. What is the weight of a 2-foot cube of gold, the specific 
gravity of gold being 19.36 ? 

LEAST COMMON MULTIPLE AND GREATEST COMMON 

DIVISOR. 

6. Find the L. C. M. of |, f , and ^-V 

7. Find the G. C. D. of |, f, f . 

8. Four bells toll at intervals of 3, 7, 12, and 14 minutes 
respectively. If they begin at the same time, how often will 
they toll together in 7 hours ? 

Ill 



112 TABLE BOOK AND TEST PROBLEMS, 

9. Four men make regular excursions into the country, 
between which each stays at home just one day. A is always 
absent 3 days, B 5 days, and C and D each 7 days. Provided 
they all start out together, how many days must elapse till 
they can all be at home at one time ? 

10. A, B, and C start at noon from the same point to travel 
around a circle of 320 rods. A walks 8, B 13, and C 24, rods 
per minute. A and B travel in one direction, and C in the 
opposite direction. How long before they all meet at the 
starting-point, no one varying his rate of travel ? 

PARTNERSHIP. 

11. A and B enagge in business as equal partners. On set- 
tlement it is found that A owes the firm $ 240, and that the 
firm owes B $260. How much should A give B to square 
the account ? 

12. Two partners, A and B, gain $249. A owns three 
fourths of the stocky lacking $ 10, and gains $ 175. Find the 
amount of their stock. 

13. A, B, and 0, having 4 loaves, for which A paid 5 cents, 
B 8 cents, and G 11 cents, eat 3 loaves, and sell the fourth to 
D for 24 cents. Divide the 24 cents equitably. 

14. Jones hires a rig for f 10 to go from Salem to Derry, a 
distance of 10 miles. At Tiffin, midway between the two 
places, he takes in Smith, who agrees to pay his proportionate 
share if Jones will take him to Derry and back again to Tiffin. 
How much should Smith pay ? 

PROPORTION. 

15. If one third of 6 were 3, what would one half of 6 be ? 
If 3 were one third of 6, what would one half of 6 be ? 

16. How long will it take 18 men to empty a tank if 15 
men can empty it in 30 minutes, and 12 men can empty it in 
40 minutes, supposing water to be running in at a uniform 
rate ? 



AEITHMETICAL PROBLEMS. 113 

17. A burden of 200 pounds suspended on a pole 4 feet in 
length, the point of suspension being 6 inches from the middle, 
is carried by two men, one at each end of the pole. How 
many pounds does each man carry ? 

18. A body weighs 1800 pounds at the surface of the earth. 
If the earth's diameter is 8000 miles, what will the body 
weigh 2000 miles above the surface ? 

19. A wins 9 games out of 12 when playing against B, and 
5 out of 8 when playing against C. Hoav many games out of 
70 should B win when playing against C ? 

20. In a pair of scales a body weighed 31^ pounds in one 
scale, and only 20 pounds in the other. Kequired the true 
weight. 

21. If 20,000 cubic feet of air per minute pass through an 
air-course 5 feet square and 1000 feet in length, what volume 
will pass through the same air-way per minute if its length be 
increased to 4000 feet, the power remaining the same ? 

22. A town clock, whose pendulum is 15 feet long, loses 3 
hours per week. How much must the ^^bob^' be raised that 
the clock may run true ? 

PROFIT AND LOSS. 

23. A pays B $ 190 for a f 240 note due in 4 years. What 
amount does A gain, money being worth 5 per cent ? 

24. If my gain is 12^ per cent of my selling price, what is 
my rate per cent of gain ? 

25. Bought 3 watches for $90, and sold them at equal 
prices. On the first I gained 80 per cent, on the second 20 
per cent, and on the third I lost 10 per cent. Find the cost 
of each. 

26. Sold a cargo of sugar at an advance of 20 per cent. 
Had it cost $ 500 more, my gain would have been only 15 per 
cent. What was the cost of the cargo ? 

ELLAVOOD^S TEST PROB. 8. 



114 TABLE BOOK AND TEST PROBLEMS. 

27. When goods are bought at 10, 10 and 5 per cent off, and 
sold at an advance of 50 per cent, at what per cent of the origi- 
nal price are they sold ? 

28. A drover bought 20 cows. and 30 oxen for $1500. He 
sold his cows for $21.60 each, his oxen for $37.50 each, and 
gained as much as one cow and one ox cost him. Find the 
price of one. 

29. I marked goods to gain 40 per cent ; but, my yard-stick 
being too long, I made only 20 per cent. Find the length of 
the yard-stick. 

30. A sells an article to B at a gain, and B to C at same 
rate of gain for $16. If B had sold for $10, his loss would 
have been half what he now gains. Find what A paid for the 
article. 

31. If I sell my sugar at a certain price per pound, I will 
lose $ 1 ; but if I increase the price 3 cents per pound, I will 
gain 50 cents. How many pounds have I ? 

32. A brewery is worth 4 per cent less than a tannery, and 
the tannery 16 per cent more than a boat. The owner of the 
boat has traded it for 75 per cent of the brewery, losing thus 
$ 103. What is the tannery worth ? 

33. I marked goods to gain 50 per cent ; but, by using an 
incorrect yard-stick, I made only 20 per cent. Find the 
length of the yard-stick. 

34. A shop-keeper buys wool from Mr. Jones, weighing it 
on false scales, and thus defrauding him to the extent of 20 
per cent. The dealer, in selling the wool, weighs it on true 
scale-s. If his entire gain on a given quantity of wool is 50 
per cent, what per cent has been gained by honest trading ? 

35. A horse which I bought for 30 per cent less than his 
real worth, having been injured, was bought from me for 25 
per cent less than he cost, and by the transaction I lost $ 55 of 
his original value. What did I get for the horse ? 



ARITHMETICAL PROBLEMS. 115 

36. A grocer sells lard at a profit of 14^ per cent. In both 
buying and selling he uses a false balance, 13 pounds in one 
scale balancing 14 pounds in the other, and thus on a given 
quantity of lard increases his profits by $ 29. Find the cost 
of the " given quantity.'^ 

37. If an article had cost me 20 per cent less, my rate of 
gain would have been 30 per cent more. Find the gain. 

38. A bought a lot for f 100 on a credit of 6 months, and 
sold it at once for $200 cash. What did he gain, money 
being worth 6 per cent ? 

39. Sold a lot for $ 100 (cost price) on a credit of 6 months, 
and bought it back at once for $ 200 cash. What did I lose, 
money being worth 6 per cent ? 

STOCKS AND BONDS. 

40. My U. S. 5's yield 7 per cent. At what discount were 
they bought ? 

41. What sum must be invested in mining stock (par $ 10) 
at 20 per cent premium, i per cent brokerage, if it pays 5 per 
cent semi-annual dividends, to yield $ 2000 yearly ? 

42. A man pays $21,200 for 5-20's when selling at 106. 
What is his annual income in currency, and per cent, gold 
being 112^ ? 

43. Which is the better investment, — U. S. 5's at 75, or 
U. S. 6's at 85 ? 

44. What rate per cent of income shall I receive if I buy 
U. S. 5's at a premium of 10 per cent, and receive pa^yment at 

par in 15 years ? (From fish's complete Arithmetic.) 

45. I made $ 5000 by a speculation, and, wishing to invest 
it permanently, I bought $2000 6's of '81 at 117f, and in- 
vested the remainder in the new 4i-'s at llOi What surplus 
remained after deducting brokerage, and what was my annual 
income ? 



116 TABLE BOOK AND TEST PROBLEMS, 

46. John Smith, through his broker, invested a certain sum 
of money in New York State 6's at 107|-, and twice as much 
in U. S. 5's of ^81 at 98|-, brokerage in each case i per cent. 
The annual income from both is $ 3348. How much did he 

invest in each kind of stock? {Yi^b's Complete Arithmetic,!^. U^.) 

INTEREST. 

47. B's fortune added to two thirds of A's, which is to B's 
as 2 to 3, being on interest for 6 years at 8 per cent, amounts 
to $ 8880. Find fortune of each. 

48. A borrows $ 100 for a year at 6 per cent interest, pay- 
ing the interest in advance. At the end of the year he pays 
$ 40 on account, and gives a new note for the balance, paying 
interest for a year in advance out of the $40. For what 
amount must the note be drawn, not reckoning days of grace ? 

49. A owes B $ 1000, but is able to raise only $ 730, with 
which he proposes to pay part of the principal and the inter- 
est in advance on the remainder. For what sum must he give 
his note at simple interest for 2 years at 5 per cent ? 

50. The present value of a freehold estate of $100 per 
annum, subject to a payment of a certain sum at the end of 
every two years, is $ 1000, allowing 5 per cent compound 
interest. Find the biennial payment. 

51. What rate per cent does a bank make on its money by 
loaning it on 90-day paper ? 

52. A man agreed to pay $6000 for a store, the principal 
and interest to be paid in three equal annual payments. Find 
the yearly payment, interest being 6 per cent. 

53. A note bearing 6 per cent interest amounted to $ 325 
May 1, 1885. It would have amounted to $ 375 Aug. 1, 1886, 
if the rate had been 8 per cent. Find face and date of note. 



ARITHMETICAL PROBLEMS. 117 

DISCOUNT AND PRESENT WORTH. 

54. Find the present worth of f 1000 due in 60 days, 
money being worth 6 per cent. 

55. Bought a lot for 1 600, of which I paid $ 50 cash, f 150 
in 9 months, $ 200 in 1 year 9 months, and $ 200 in 2 years 9 
months, interest at 8 per cent. What was the cash value, 
money being worth 6 per cent ? 

56. Bought a horse for $ 500, and immediately sold him on 
a credit of 6 months. The First National Bank discounted 
the note I received ; and, having examined my money, I found 
I had gained 20 per cent on my purchase. What was the face 
of the note ? 

57. At what rate must a note, payable in 60 days, be dis- 
counted to produce 6 per cent interest ? 

58. What must be the face of a note that yields $2000 
when discounted at bank the day of its date, if drawn 30 days 
after date ? 

59. A note is payable in 30 days. At what rate must it be 
discounted to produce 6 per cent interest ? 

60. A bank by discounting a note at 6 per cent receives for 
its money a discount equivalent to 6^ per cent interest. How 
long must the note have been discounted before it was due ? 

61. A man bequeathed $9000 to his three sons, aged 13, 
15, and 17 years, in such a manner that the share of each, 
placed at compound interest at 6 per cent until he arrived at 
the age of 21 years, should amount to the same sum. Find 
the share of each. 

INVOLUTION AND EVOLUTION. 

62. Extract the square root of .625. 

63. Find the number when two thirds of its cube is 10 more 
than the cube of its two thirds. 



118 TABLE BOOK AND TEST PROBLEMS. 

64. Two thirds of the square of three fourths of a number 
is 12 more than three fourths of the square of one half the 
number. What is the number ? 

65. What must be the width of a walk around a park 40 
rods square, to contain one fourth the area of the park ? 

ALLIGATION. 

66. Sold 50 dozen eggs for f 8. How many dozens of each 
were there at 13, 14, 18, and 21 cents respectively ? 

67. What relative quantities of gold and silver, whose spe- 
cific gravities are 19^ and 10|-, will make a compound whose 
specific gravity shall be 16.84 ? 

ANNUITIES. 

68. I rent a house for $300 a year, the rent to be paid 
monthly in advance. What amount of cash at the beginning 
of the year will pay one year's rent ? 

69. B bought a house for $6000 down, or equal install- 
ments of $ 1200 a year for 6 years. Which is the better for 

B, money being worth 6 per cent ? (From Brooks's Arithmetic.) 

70. A man borrows $ 1000 at 6 per cent per annum, to be 
repaid in five equal annual payments. How much must be 
paid each year, interest being added to principal at end of 
each year, and before the annual payment is deducted ? 

"AGE" PROBLEMS. 

71. My age now is one fourth of yours, but in 20 years I 
will be one half as old as you. How old are we ? 

72. Ten years ago, when Mrs. C. was married, she was one 
third as old as her husband, but now she is three sevenths as 
old. How old was each when they were married ? 



ARITHMETICAL PROBLEMS, 119 

73. Ten years ago Edwin was one third as old as his father, 
but 2 years hence he will be one half as old. What is the age 
of each ? 

74. Three times Jennie's age equals three eighths of Gertie's 
age. In how many years will Gertie be just twice as old as 
Jennie ? 

"TIME" PROBLEMS. 

75. What is the time if one fifth of the time past noon 
equals one third of the time to noon again ? 

76. Three fourths of the time past noon equals one half the 
time to midnight lacking 1 hour. What time is it ? 

77. It is between 4 and 5 o'clock, and the hands on the dial 
are exactly opposite. What time is it ? 

78. The hour hand of a clock moves 10 per cent too fast, 
and the minute hand 5 per cent too slow. In 20 minutes 
(true time) they will be together. How many minute spaces 
are they apart now ? 

79. At what times between 6 and 7 o'clock are the hour 
hand and minute hand 20 minutes apart ? 

80. At a certain time between 1 and 2 o'clock the minute 
hand was between 2 and 3. W^ithin an hour the hands had 
changed places. What was the time when the hands were in 
the first position ? 

81. It is between 5 and 6 o'clock, and the minute hand has 
passed the 6 as far as the hour hand lacks having reached it. 
What time is it ? 

82. At a certain time between 2 and 3 o'clock the minute 
hand was between 3 and 4. Within an hour after, the hour 
hand and minute hand had e-xactly changed places. . What was 
the time when the hands were in the first position ? 

83. A clock has three hands, — hour, minute, and second, — 
all turning on the same center. At 12 o'clock all are together, 



120 TABLE BOOK AN B TEST PBOBLEMS. 

and point at 12. How long will it be before each hand will be 
midway between the other two ? 

GENERAL ANALYSIS. 

84. If 48 pounds of sea-water contain 1|- pounds of salt, 
how much fresh water must be added to the 48 pounds so that 
40 pounds of the mixture shall contain one half a pound of 
salt ? 

85. Suppose A and B work together for 4 hours, after which 
A leaves, and B finishes in 3 hours 36 minutes ; but if B had 
left, A would have done the remainder in 4 hours 30 minutes. 
In what time could each alone have done the work ? 

86. A, B, and C together have f 150. If B's money were 
taken from the sum of the other two, the remainder would be 
$ 50 ; and if C's were taken from the sum of the other two, 
the remainder would be one half of his (C's) money. Find 
how much each had. 

87. When a certain number of men had labored 12 days, 
and had the job one third completed, 20 more men were set to 
work, and the remainder of the job was finished in 14 days. 
How many men were employed at first ? 

88. A tree 33 feet high breaks off. If the broken part 
were a foot longer and the stump 4 feet shorter, the stump 
would be twice as long as the broken part. Find the length 
of both parts. 

89. A and B can do a piece of work in 15 days, B and C in 
10 days, and A and C in 12 days. How long will it take all 
to do the work ? 

90. Bought 10 bushels potatoes and 20 bushels of apples 
for f 11. At another time I bought 20 bushels potatoes and 
10 bushels of apples for $13. What price per bushel was 
paid? 



ARITHMETICAL PBOBLEMS. 121 

91. A, B, and C each, contemplated purchasing a warehouse. 
They agreed that if A and B bought it^ A should pay two thirds 
of the price ; but if B and C bought it^ B should pay two 
thirds of the price. They finally agreed to buy it together, 
and it was found that A paid $ 12;000 more than C. Find 
the cost. 

92. Two boats leave the same shore at the same time. It 
takes one 12 minutes to reach the opposite shore, and the 
other sails three times as fast. When will they first meet if 
the fast boat stops 2 minutes at the opposite shore ? 

93. How many tucks one fourth of an inch wide can be 
made in a strip of muslin a yard long, leaving one eighth of an 
inch between the edge of one tuck and stitching of the next ? 

94. I have two fifths of my money stolen. I then earn 
$ 60, and spend two thirds of all I now have. My uncle then 
gives me $10, and after losing one sixth of what I have I find 
I have half as much as I had at first. How much had I 
at first ? 

95. A tree 66 feet high breaks off. If the broken part 
were 2 feet longer and the stump 8 feet shorter, the former 
would be half as long as the latter. Find the length of the 
stump. 

96. A man bought a cow, a horse, and an ox for $350. 

The horse cost 4 times as much as the ox lacking $ 40, and 
the ox cost twice as much as the cow lacking $ 12. Find the 
cost of each. 

97. I can pick 40 bushels of potatoes or dig 20 bushels in a 
day. How many bushels can I dig and pick in a week ? 

98. If a man can pasture 2 cows on an acre, and allows 
1 acre of roots to every 6 cows, how many cows can he keep on 
20 acres, and how much land has he in grass ? 

99. The head of a fish is 3 inches long ; the tail is as long 
as the head plus one fourth of the body ; and the body is as 
long as the head and tail. How long is the fish ? 



122 TABLE BOOK AND TEST PROBLEMS. 

100. A watch, aud diain cost $ 100. Half of the cost of tlie 
watch is f 20 more than th.e cost of the chain. Find the cost 
of each. 

101. A beat B by 80 feet in a foot-race; but if A had 
walked eight ninths of his rate, and B nine tenths of his, A 
would have beaten B by only 26 feet. Find the length of the 
course. 

102. Equal weights of gold and silver are in value as 20 to 
1, and equal volumes are in value as 1284 to 35. A certain 
volume is composed of equal weights of gold and silver. Find 
how many times more valuable it would be were it composed 
of gold alone. 

103. A and B engaged to reap a field for 90 shillings. A 
could reap it in 9 days, and they promised to complete it in 
5 days. They found, however, that they were obliged to call 
in C, an inferior workman, to assist them the last two days, 
in consequence of which B received 3s. 9d. less than he other- 
wise would. In what time could B and C each reap the field ? 

(From Mathematical Magazine.) 

104. The product of two numbers is 117, and their quo- 
tient is 1.4. Find the numbers. 

105. Two trains start at the same time, one from Pittsburgh 
to Altoona, the other from Altoona to Pittsburgh. If they 
arrive at destinations 1 hour and 4 hours after passing, what 
are their relative rates of running ? 

MENSURATION. 

The Rectangle, 

106. The perimeter of a rectangular field is 200 rods, and 
three times the width is twice the length. Find the area in 
acres. 

107. The sides of a rectangular room are as 4 to 3. If each 
side were 2 feet longer, the area would be 1344 square feet. 
Find the sides. 



ARITHMETICAL PROBLEMS. 123 

108. A rectangular trough is two thirds full of water. After 
35 gallons are taken out, it is three eighths full. What is the 
depth, the length being 10 feet and the width 2| feet ? 

109. Around a garden 80 feet square is a walk containing 
one sixth as much area as the garden. Find the width of 
the walk. 

110. The largest square stick that I can cut from a 30-foot 
log contains 45 cubic feet. Find its diameter. 

111. Find the area of a square field whose diagonal is 8.284 
rods longer than its side. 

112. From one corner of a rectangular pyramid 6 by 8 feet 
it is 13 feet to the apex. Find the dimensions of a rectangular 
solid whose dimensions are as 2, 3, and 4, that may be equiv- 
alent in volume. 

113. Two men run in opposite directions around a rectangu- 
lar field whose area is 1 acre. They start from one corner, 
and meet 13 yards from the opposite corner. The rates of 
running being as 5 to 6, find the dimensions of the field. 

114. The inside dimensions of two similar rectangular oil- 
cans are as 3, 7, and 11. The first holds 8 gallons, and the 
second requires 4 times as much tin as the other. Find the 
dimensions of the smaller, and the volume of the larger. 

The' Triangle. 

115. A doorway is ,8 feet high, and just wide enough to 
allow a circular saw 10 feet in diameter to pass through. 
Find the width. 

116. An isosceles triangle contains 6 acres and 12 perches, 
and its base is 72 perches. Find the sides. 

117. Find the area of a triangle whose sides are 9, 14, and 
5 rods. 

118. A man has a triangular field whose sides are 40, 60, 
and 60 rods respectively. A fence is built from the middle 



124 TABLE BOOK AND TEST PROBLEMS. 

point of the side 50 rods long to the middle of the side 60 rods 
long. How long is the fence ? 

119. In a two-thirds pitch roof what is the length of the 
rafters if the building is 36 feet wide ? 

120. How long a rope will wind once around a cylinder 10 
feet long and 6 feet in diameter^ commencing at one end and 
going spirally around to the other ? 

121. A level lot of land 300 feet square has a wall 9 feet 
high surrounding it. What is the least height from the center 
of the ground that a man must stand who measures 5 feet to 
his sights so that by looking over one corner he may see an 
object on the ground 20 feet distant from it ? 

(From Roybr's Problems.) 

122. A triangular meadow whose sides each measure 100 
rods has a horse tied at one vertex. How long a rope will 
allow him to graze over half the meadow ? 

123. In a cubical room a line drawn from an upper corner 
to the middle of the floor is 24 feet. What is the size of the 
room ? 

124. Two trees stand on opposite sides of a stream 40 feet 
wide. The height of one tree is to the width of the stream as 
8 to 4, and the width of the stream is to the height of the other 
as 4 to 5. What is the distance between their tops ? 

125. A room is 12 feet wide, 16 feet long, and 8 feet high. 
A fly wishes to crawl from a lower corner to an opposite upper 
corner by the shortest route. How far must it travel ? 

126. How many feet of boards can be sawed from a log 
whose diameter is 2V2 feet, and length 16 feet, allowing one 
fourth of an inch as the width of saw ? 

127. A trapezoidal board, 12 feet long, is 16 inches wide at 
one end, and 8 inches at the other. How far from either end 
must it be cut transversely so that each part may contain one 
half of it ? 



ARITHMETICAL PROBLEMS. 125 

The Circle. 

128. A circular floor 30 feet in diameter is surrounded by 
a granary 4 feet wide and 3 feet high. Find the contents of 
the granary. 

129. If my plow cuts 18 inches wide, how many times must 
I plow around a circular quarter section to plow one half 
of it ? 

130. In turning a sulky, whose wheels are 5 feet high and 
6 feet apart, so that the outer wheel is kept on the circumfer- 
ence of a certain circle, it is observed that the outer wheel 
makes two revolutions while the inner wheel makes one. 
Find the circumference of the inner track. 

131. If a man 6 feet high should walk once around the 
earth on the equator, how much farther would the top of his 
head move than his feet ? "* ^ 

132. A horse is tied to one corner of a square 10-acre field. 
How long must a rope be to allow him to graze over 10 acres 
outside the field ? 

133. A, B, and C bought a grindstone 3 feet in diameter 
and 3 inches thick for f 5. A paid $2; B, f 1.75; and C, 
f 1.25. How many inches must each grind off to get the 
worth of his money ? 

134. The length of the longest straight line that can be 
drawn on the surface of a circular race track is 20 rods. What 
is the area of the track, and what its width ? 

Pyramids and Cones. 

135. How many square yards of cloth will be required to 
make a conical tent 10 feet in diameter and 12 feet high ? 

136. Three men bought a conical sugar-loaf 20 inches high, 
and divided it into three equal solids by sections parallel with 
the base. Find the height of each section. 



126 TABLE BOOK AND TEST PROBLEMS. 

137. A conical vessel, whose altitude is 12 inches, contains 
13 gallons of water, and the area of the bottom is to that of 
the top as 5 to 3. Find the two dimensions. 

138. The diameters of the frustum of a cone are 40 and 20 
feet, and the altitude 60 feet. What length must be taken 
from the larger end to contain 12,000 cubic feet ? 

139. The volume of the frustum of a cone is 7050 cubic 
inches, its altitude 12 inches, and the diameter of its lower 
base twice that of the upper base. Find the diameters of the 
two bases. 

140. A small bucket, one third full, is 8 inches deep, and 
its upper and lower diameters are respectively 7 and 6 inches. 
Find the volume of a ball, which, falling in, would cause a rise 
of 3 inches. 

141. Find the volume of the largest square pyramid that 
can be cut from a cone whose diameter is 10 feet, and altitude 
30 feet. 

Similar Solids. 

142. A bushel measure is 18^ inches wide and 8 inches deep. 
What are the dimensions of a similar measure containing 8 
bushels ? 

143. A conical wineglass 2 inches in diameter and 3 inches 
deep is one fourth full of water. Find depth of water. 

144. Goliath of Gath weighed 1015 pounds. What was his 
height if a man 5 feet 10 inches in height weigh 180 pounds ? 

145. There are three balls whose diameters are 3, 4, and 5 
inches respectively. What is the diameter of a ball contain- 
ing as much as the three ? 

146. A solid metal ball has a radius of 4 inches, and weighs 
8 pounds. AVhat is the thickness of a spherical shell of the 
same metal, weighing 7|- pounds, the external diameter of 
which is 10 inches ? 



ARITHMETICAL PROBLEMS, 127 

Cubes and Spheres. 

147. How many gallons of water in a hollow sphere, the 
diameter of which is 12 inches, and the crust 1 inch thick ? 

148. A cube immersed in a rectanglar reservoir 36 inches 
long and 16 inches wide raises the water 3 inches. What is 
the edge of the cube ? 

149. Eind the diameter of a metal sphere which, dropped 
into a 6-inch cylindrical vessel, raises the water 4 inches. 

150. A cylindrical bucket 10 inches in diameter is one third 
full of water. A ball dropped into it raises the water to the 
brim. Find the depth of the bucket if the ball is just sub- 
merged. 

151. The greatest cube that can be inscribed in a given 
sphere has 10 inches for its diagonal. What are the contents 
of that portion of the sphere between its surface and the cube ? 

152. Four ladies own a ball of fine thread 3 inches in diam- 
eter. What portion of the diameter must each wind off in 
order to have equal shares of thread ? 

153. A conical wineglass which is brimful measures across 
the mouth 6 inches, and in depth 8 inches. What amount of 
water will run over if a sphere 4 inches in diameter be put 

into it ? (From Spiegel's Live Questions.) 



MISCELLANEOUS PROBLEMS. 

154. In division of fractions why do we invert the divisor 
and multiply ? 

155. Find the value of 15 + 9 -f- 3 - 2 x 3. 

156. A dry oak log is 20 feet long, 3 feet wide, and 2^ feet 
thick. What is its weight, the specific gravity of dry oak 
being .925 ? 



128 TABLE BOOK AND TEST PROBLEMS. 

157. The probability that Ulerich can lift 300 pounds is two 
thirds : the probability that Bierer can lift the same is five 
twelfths. If both try, what are the probabilities (1) that U. 
lifts it and B. fails ; (2) that U. fails and B. lifts it ; (3) that 
both lift it ; (4) that neither lifts it ? 

168. The population of a State is 6,000,000. One seventieth 
die yearly. One sixtieth are born annually. What will be 
the population in 300 years ? 

159. A took 50, B 30, and C 10, eggs to market. They sold 
at the same price, and all received the same amount of money 
for their eggs. How was that possible ? 

160. Two trains, one 210 feet long, and the other 230, move 
on parallel tracks. When going in the same direction, they 
pass each other in 15 seconds ; and when going in opposite 
directions, they pass in 3| seconds. Find the rates of the 
trains. 

161. What number divided by 11 leaves a remainder of 9, 
divided by 9 leaves a remainder of 6, by 7 leaves a remainder 
of 5, by 6 leaves a remainder of 4 ? 

162. From a cask of wine containing 100 gallons, 10 gallons 
are drawn, and the cask filled up with water ; 10 gallons are 
again drawn, and the cask filled. This process is repeated until 
100 gallons have been drawn from the cask. How much wine 
remains ? 

163. A postman delivered daily for a period of 42 days 4 
letters more than on the previous day. The aggregate delivery 
for the last 18 days was the same as that of the first 24 days. 
How many letters did he deliver altogether ? 

164. Three men, named Smith, Brown, and Hugus, with 
their sons Adam, Paul, and Dick, each own a piece of land in 
square form. Mr. Smith's piece is 23 rods longer on each side 
than Paul's, and Mr. Brown's piece is 11 rods longer on each 
side than Adam's. Each man has 63 square rods more than 
his son. What are the full names of the boys ? 



ABITHMETICAL PROBLEMS. 129 

165. If 3 acres of grass, together with what grew on the 3 
acres while they were grazing, keep 12 oxen 4 weeks, and in 
the same manner 5 acres keep 15 oxen 6 weeks, how many- 
oxen can in the same manner graze on 6 acres for 9 weeks ? 

166. Eequired the greatest possible number of hills of corn 
that can be planted on a square acre, the hills to occupy only 
a mathematical point, and no two hills to be nearer than 3^ 
feet. 

167. How many trees can be planted on a small field 11 
rods square, no two trees to be nearer to each other than 1 
rod? 

168. Arrange the figures 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, in such a 
way, that, when added, the sum will be just 100. 



9. 



130 TABLE BOOK AND TEST PROBLEMS. 



ALGEBRAIC PROBLEMS. 
FACTORING. 

169. Factor - 16 ± 40VS-25a?. 

170. Factor x'^x-^O. 

171. Factor x^ -2x'' -2>x + &. 

172. Resolve x into 3 unequal factors. 

173. Factor ar^ — aV — aV + a;?/". 

174. Factor i»2 + 10x-39. 

175. Factor x'-bx-lL 

176. Factor 10 a (~ + ^V 20 a. 

177. Factor x^ — y'^ and oi^ + y'^. 



x + 



FRACTIONS. 

a 



1+^ 

178. Simplify -• 

X— - 



1-5 



a 



179. What is the reciprocal of ) rf— ? 

^ {a + by 

180. Free ^^""^-^ of negative exponents. 

x'^ + y ^ 



ALGEBRAIC PROBLEMS. 131 

SIMPLE EQUATIONS. 

181. Given x + y=a (1) 

x + z=b ■ (2) 

y + z = c (3) 

182. Divide the fraction f into two parts, so that the 
numerators of the two parts taken together shall be equal to 
their denominators taken together. 

183. A person engaged to work a days on these conditions : 
for each day he worked he was to receive b cents, and for each 
day he was idle he was to forfeit c cents. At the end of a 
days he received d cents. How many days was he idle ? 

184. A man can row with the current from A to B, a dis- 
tance of 42 miles, in 3 hours. When the current is two thirds 
as strong, it takes him 10^ hours to row from B to A. What 
was the velocity of the current in the first case ? 

185. A banker has two kinds of change. There must be a 
pieces of the first to make a crown, and & pieces of the second 
to make the same. E'ow, a person wishes to have c pieces for 
a crown. How many pieces of each kind must the banker 
give him ? 

186. A number is represented by 6 digits, of which the 
left-hand digit is 1. If the 1 be removed to units^ place, 
the others remaining in the same order as before, the new 
number will be 3 times the original number. Find the number. 

187. The figure 7 is exactly midway between the hour and 
minute hands, and it is between 6 and 7 o'clock. What is the 
time? 

188. In an alloy of silver and copper, — of the whole +p 

H m 

ounces was silver, and - of the whole — q ounces was copper. 

n 

How many ounces of each were there ? 



132 TABLE BOOK AND TEST PROBLEMS. 

189. A and B ran a mile, A giving B a start of 20 yards at 
the first heat, and beating him 30 seconds. At the second 
heat A gives B a start of 32 seconds, and beats him Oy^y yards. 
At what rate per hour does A run ? 

190. A certain article of consumption is subject to a duty 
of 6 cents per pound ; but, in consequence of a reduction of 
the duty, the consumption increased one half, but the revenue 
fell one third. What was the duty per pound after the reduc- 
tion? 

191. It is between 3 and 4 o'clock, and the hour hand is 
exactly opposite the minute hand. What time is it ? 

192. Find four numbers, such that the first plus half the 
rest, the second plus a third the rest, the third plus a fourth 
the rest, and the fourth plus a fifth of the rest, shall each be 
equal to a. 

193. A person has just 2 hours' spare time. How far may 
he ride in a stage which travels 12 miles an hour, so as to 
return home in time, walking back at the rate of 4 miles an 
hour? 

194. A vessel sailed with the wind and tide 60 miles, and 
returned with the wind and against the tide. She reached the 
same point in 12 hours, and the rates of sailing out and in 
were as 5 to 3. Kequired the time each way, and the strength 
of the wind and tide. 

195. Two years ago Mr. Jones was 6 times as old as his son 
John will be 2 years hence, and 3 years hence his age will 
be 15 times John's age 3 years ago. How old is each ? 

196. At 12 o'clock the three hands of a clock are exactly 
together, all moving on the same center. How long before 
each hand will be exactly midway between the other two ? 

197. 1^ ? f f f ? ^^ A, B, and C start at 
the same time from R to travel 40 miles to W. A walks at 
the rate of 1 mile an hour, B 2 miles an hour, while C rides 8 



ALGEBRAIC PROBLEMS. 133 

miles an hour. C rides to W, and then back until he meets A, 
whom he picks up and carries a certain distance, then again 
rides back and picks up B, whom he carries just far enough to 
allow all three to reach W at the same time. Find the time 
of the trip. 



RADICALS. 



198. 



From (a — x) ^o? — oi? subtract (a — ^)\/ ^ ^ ' 
199. Extract the square root of (1 — a;^)"^ + 1. 



200. Prove V^^^^^Vx-Va. 



201. Find the square root of \w? + |-7iVm- — v?. 

QUADRATIC EQUATIONS. 

202. A tree a feet in height stands at the edge of a stream 
h feet in width. Where must this tree break so that the top 
may reach across the stream, while the broken parts remain in 
contact ? 

203. What is the diameter of a sphere which contains as 
many cubic inches as there are square inches in its surface ? 

204. Given x + y=za (1) 

xy^h (2) 

to find X and y, 

205. Given V^-V^=2V^ (1) 

0^ + 2/= 20 (2) 

to find X and y, 

206. If a number be increased by 6, and then the same num- 
ber be decreased by 6, the difference between the square roots 
of the results is 6. What is the number ? 

207. Given a^ + l = 0, 

or 

to find X. 



134 TABLE BOOK AND TEST PBOBLEMS. 

208. From a cask containing 81 gallons of wine^ a man 
draws off a certain quantity, and then, filling the cask with 
water, draws off the same quantity again, and then there 
remain only 36 gallons of pure wine. How much wine did 
he draw off each time ? (To be solved by pure quadratics.) 

209. A picture is 10 by 16 inches, and the surface of its 
frame is equal to the surface of the picture. Find the width 
of the frame. 

210. The number of rods around a square field is equal to 
the number of acres in it. Find the area. 

-\/ax + Vn Va + Vn 



211. Given 
to find X. 



-\/ax — ■\Jn '\fn 



212. Given cc^ — (a — 5 + c) a; = (6 — a) c, 
to find X, 

213. An army, 25 miles from front to rear, moved forward 
just the length of itself. When it commenced to move, an 
officer started from the rear, rode to the front, and then back 
to the rear, which he reached just as the army halted. What 
distance did the officer travel ? 

214. The loudness of one bell is 3 times that of another. 
Now, supposing the strength of sound to be inversely as the 
square of the distance, at what place on the line of the two 
will the bells be equally well heard, the distance between them 
being a ? 

215. The intensity of two lights, A and B, is as 7 to 17, and 
their distance apart 132 feet. Assuming that the intensity 
varies inversely as the square of the distance, where in the 
line of the lights are the points of equal illumination ? 

216. What price are eggs per dozen, when to give 2 more 
for 12 cents would decrease the price x^er dozen 1 cent ? 



217. Given Jl + - = Jl _ ^ + 1, to find x. 



ALGEBBAIC PROBLEMS. 135 

218. A and B husk a field of corn for $20. When they 
husk 5 rows at a time, A husks 3 standing rows, while B husks 
the ^^down^' row and 1 standing row. When they husk 6 
rows at a time, B husks 3 standing rows, while A husks the 
"down'^ row and the other 2 standing rows. How must 
they share the money ? Two elevenths of the rows are 
" down '^ rows : what is it worth to husk them ? 

219. Given a;^" - 2 i^'* + aj" = 6, to find x, 

220. There is a number consisting of two digits, which 
being multiplied by the left-hand digit gives the product 46 ; 
but if the sum of the digits be multiplied by the same digit, the 
product is only 10. What is the number ? 

221. Eind two numbers such that their sum, their product, 
and the difference of their squares, shall be all equal to each 
other. 

222. Given -^— = n, to find x. 

■\/a + V a — Va^ — ax 

(From Stoddard and Henkle's Algebra.) 

223. The sum of three numbers in harmonical proportion is 
191, and the product of the first and third is 4032. Eind the 
numbers. 

224. How large a trough can be dug out of a square stick of 
timber twice as long as wide and deep ; the sides, end, and 
bottom to be 3 inches in thickness ; and when completed to 
contain exactly 11,772 solid inches of timber ? 

225. Find three numbers such that if the first be multiplied 
by the sum of the second and third, the second by the sum of 
the first and third, and the third by the sum of the first and 
second, the products shall be respectively 2&, 60, and 56. 

226. A and B engaged to reap a field for 90 shillings. A 
could reap it in 9 days, and they promised to complete it in 
5 days. They found, however, that they were obliged to call 



136 TABLE BOOK AND TEST PROBLEMS. 

in C, an inferior workman, to assist them the last 2 days, in 
consequence of which B received 3s. 9d. less than he other- 
wise would. In what time could B and C each reap the field ? 

227. A railroad train, after traveling for 1 hour, meets 
with an accident which delays it 1 hour, after which it pro- 
ceeds at three fifths of its former rate, and arrives at the 
terminus 3 hours behind time. Had the accident occurred 60 
miles farther on, the train would have arrived 1 hour and 
20 minutes sooner. Eequired the length of the line, and the 

original rate of the train. (From Todhunter's Algebra.) 

228. The sum of four numbers in geometrical progression 
is 15, and the sum of their squares 85. Find the numbers. 

229. The product of two numbers is p, and the difference of 
their cubes is equal to m times the cube of their difference. 
Find the numbers. 



SPECIAL EXPEDIENTS. 137 



SPECIAL EXPEDIENTS. 
SIMULTANEOUS EQUATIONS. 

230. Givea x' + y =11 (1) 

x+f = l (2) 

to find a; and 2/ by quadratics. 

231. Given ^^^rs + s^^m^ (1) 

s^ + st-\-f = n2 . (2) 

^2 + r^+r2 = p' (3) 

to find r, s, and t. 

232. Given x + y = 10 (1) 

icV^=12 (2) 

to find X, 

'233. Given x' + y' = 34. (1) 

^2/ = 15 (2) 

to find X and y, . 

234. Given a; + 2/ = 0^2 (-L) 

32/-a; = 2/' (2) 

to find X and y. 

235. Given x +y =4. (1) 

^^ + 2/^=82 (2) 

to find ic and y, 

236. Given a; - ?/ = l (1) 

x^ _ 2/3 = 19 (2) 

to find X and y. 

237. Given oj _fi/ + a:?/(a; + ?/) + a^y = 85 (1) 

xy + {x + yy + xy(x + y) = 97 (2) 

to find X and y. 



138 TAB. 


LE BOOK 

( ^"^ ¥4 

V^ + 2// 


ANB TEST PROBLEMS. 


, 


238. Given 


\ 3a; 


(1) 






xy- 


-(aj + 2/) = 54 


(2) 


to find X and y. 






% 




239. Given 






aj2 + a;?/ = 104 
aj2 + ^2^89 


(1) 
(2) 


to find X and y. 










240. Given 






(x + y) = 70 


(1) 

(2) 


to find X and y. 










241. Given 






X + ?/ = 5 

oc^ + y^ = &5 


(1) 
(2) 


to find X and y. 










242. Given 






ir2 + 2/2 = 34 
a;^ — a;?/ = 10 


(1) 
(2) 


to find X and y. 











243. Find two numbers whose product is equal to the differ- 
ence of their squares, and the sum of whose squares is equal 
to the difference of their cubes. 

244. What two numbers are those whose difference multi- 
plied by the difference of their squares is 32, and whose sum 
multiplied by the sum of their squares is 272 ? 

/ 245. Given 2{x + yy+l = {x'' + y^){xy + cc^ + f) (1) 

x + y = 3 (2) 

to find X and y. 

246. Given x^ + y* = 3x (1) 

x^ + y'^>=x (2) 

to find X and y. 



SPECIAL EXPEDIENTS. 139 

247. Given xy — x^ — y^ (1) 

^ + f = ^-f (2) 

to find X and y. 

248. Given {x^-y){l-\-xy) = rixy (1) 

(x^ + f) (1 + a;y ) = 208 ^f (2) 
to find X and 2/. 

249. Given x - y = 2 (1) 

a^-2/' = 242 (2) 
to find a; and y. 



250. Given 22/2- 8V^ + 2Va; V2/'-4Va; = |a; (1) 

V5 + V8(2/-V^-4) = 2/ + l (2) 

to find a; and ?/. 

251. Given ^(;2^4^(? + ^ = 6 (1) 

%^^2z-\-w=^^ (2) 

to find w and 2;. 

RECIPROCAL OR RECURRING EQUATIONS. 

252. Given {:^ -\-V)(o? + V){x + \)^?>^^, 
to find X, 

253. Given Sa;^- 16a;^- 25a.^ - 16a^ + 8 = 0, 
to find X, 



254. Given Joj -1- Jl_ 1 = ^Zl1, 

\ a; \ a? x 
to find a:. 

255. Given 1 + a;^ = a(l + aj)^ 
to find a;. 



256. Given 2x-sJ\ — x' ^ a(\ -\- x^), 

to find X, 



140 tabl:e book and test pboblems. 

257. Given 
to find all the real and imaginary values of x, 

HIGHER EQUATIONS. 



258. Given y/x' -a' + Vc^^^ = Va?' - c^ + Vx' - d% 
to find X. 

259. Given 2x^/1^^' == m{2 --x'), 
to find X, 



260. Given 2x-\/l^x' = 2--x', 
to find X, 

261. Given •\/x--\/x = 100, 
to find X. 

262. Given Sx^ + 16.'r = 9, 
to find X. 

263. Given a:3 + 2aj2 + ii;= 18, 
to find X. 



264. Given 
to find X, 






+ a;\l + a; 12 



2 a; 



265. Given x\x + 3) = 2{3x + 4.), 

to find a;. 

266. Given oj^- 6a.^ + lla? = 6, 
to find 0?. 

267. Given o;^- 6a:^ + 5a;^ + 12a; = 60, 
to find X, 

268. Given oj^ + 16 a; = 128, 
to find X, 



SPECIAL EXPEDIENTS. 141 

269. Given a^-V^=14, 
to find X, 

270. Given 2:x?^:x? = l, 
to find X, 

271. Given a;3_6aj + 4 = 0, 
to find X, 

272. Given a^- 3x^ + 4 = 0, 
to find X. 

273. Given VS-V^ = 100, 
to find X, 

274. Given 8 a;* - 20 a;^ + 20 o;^ - 60 a; = 108, 
to find X. 

275. Given 0^ + 0.^ = 80, 
to find X, 

276. Given -,--\/a.'- 2a -^ = 1, 
to find a;. 

277. Given -^|^±5^2 = oj - 2, 
to find X, 

o^Q n- (1 + ^)' , (1-^)' 

278. Given ^ ~\ i +^^ ^=^, 

1 + a;^ 1 — a;^ 

to find X. 

279. Given a;(V^ + 1)' = 32(a;+ V^)- 240, 
to find X, 

280. Given V^-— J— = -, 

Va;-2 »' 
to find a: by quadratics. 



142 TABLE BOOK AND TEST PROBLEMS. 



281. Given 


yja 


-^^F|-■' 




to find X. 








282. Given 




(!^ + l)y = (f + l)a^ 


(1) 


to find X and y. 




{f + l)x = (x' + l)9f 


(2) 


283. Given 


</2x 
</2x 


: + 3_^ ^2a;-3_ 8 /ix^ + 9\ 






:-3 ^2x + 3 13(^4a;2-9/ 




to find X. 








284. Given ; 


^(l + l) 3«. .= 70, 




to find X. 


\ 






285. Given 




l + a? = a{l + x)% 




to find X. 








286. Given 


a;3 


-6a;^ + lla;-6 = 0, 




to find X. 








287. Given 




12 + 8 VS 




to find X, 








288. Given 




a?' + 2/S = 72 


(1) 


to find X and y. 




«2/(« + 2/) = ^8 


(2) 


289. Given 




a^ + f = 35 


(1) 


to find X and y. 




a;2 + 2/2 = 13 


(2) 


290. Given 




(a;_2/)(a^_2/2) = 160 


(1) 


to find X and y. 




(a; + 2/)(a;^ + 2/2) = 580 


(2) 


291. Given 




xyz{x + y — z) = a = 24, 


(1) 






xyz{x — y + z) = b =72 


(2) 






xyz(— x + y + z) = c = 120 


(3) 



to find Xj y, and z. 



SPECIAL EXPEDIENTS. 143 

292. Solve the following equation by quadratics : — 

VpT) + VFf ) = VFf) WF^- 

293. Solve cc^- 3x^ + 4 = 0. 

294. Solve a^-6a; + 4 = 0. 

295. Solve a^-8a;' + 19x-12 = 0. 



I 

144 TABLE BOOK AND TEST PBOBLEMS. 



MISCELLANEOUS PROBLEMS. 
APPLICATIONS OF ALGEBRA. 

296. The . perimeter of a rectangle is 140 feet. Prom one 
corner to the center is 25 feet. Find dimensions and area. 

297. In a right-angled triangle are given the difference 
between the base and perpendicular, and also the difference 
between the base and hypothenuse, to find the sides. 

298. A room is 40 feet long and 13 feet wide. What is the 
length of the longest piece of carpet that can be laid on the 
floor of said room ? 

299. What is the area of a circular field in which the num- 
ber of acres is equal to the number of boards inclosing it, the 
fence being 5 boards high, and each board 1 rod long ? 

300. I have a board whose surface contains 49|- square feet. 
The board is 1^ inches thick, and I wish to make a cubical 
box of it. Eequired the length of one of its equal sides. 

301. Two poles stand on the same plane. From the top of 
the shorter to the foot of the longer is 40 feet, and from the 
top of the longer to the foot of the shorter is 60 feet. These 
two lines intersect 15 feet from the plane. Find the height 
of each pole. 

302. Draw a right line parallel to the base of a triangle, so 
that the parallel shall be equal to the difference of the lower 
segments. 

303. A pole standing perpendicularly on a hillside was 
broken by a storm a feet from the top, which touched the 
ground up the hill b feet from the foot of the pole. The mend- 
ing shortened it so that the broken part was exactly the length 
of the stump ; but it broke again at the same place, when it 



MISCELLANEOUS PROBLEMS. 



145 



was observed that the top touclied the ground c feet nearer the 
foot of the pole than before. Find original height of the pole. 

304. The hypothenuse of a right triangle is 35, and the side 
of the inscribed square is 12. Find the sides. 

305. A square farm contains as many acres as there are 
boards in the fence inclosing it. The fence is 7 boards high, 
and each board is half a rod long. How many acres are there 
in the farm? 

306. A solid globe 1 foot in diameter has been blown into 
a hoUow sphere one eighth of an inch thick. Find diameter 
of the hollow sphere. 

307. A circle is inscribed in an isosceles triangle whose 
base is twice its altitude. Show that the radius of the circle, 
divided by the altitude of the triangle, is the V2 — 1. 

308. The area of a triangle is a rods, one side is h rods, and 
the other two sides are to each other as 2 to 3. Find them. 

309. Given AB to be a 
straight line, AO = 60, 
(7i>=20, the angle ECD 
to be the complement of 
the angle Ay the angle 
EDB to be twice the angle 
A, and EB to be a perpen- 
dicular to ABy to find EB, 

310. A tree 74 feet high, standing perpendicularly on a hill- 
side, was broken by the wind, but not severed ; and the top 
fell down the hill, striking the ground 34 feet from the root of 
the tree, the horizontal distance from the root to the fallen 
part being 18 feet. At what height did it break ? 

311. From a point within a square, three lines are drawn to 
three corners of the square. Find a side of the square. 

312. The sides of an equilateral triangle are 200 feet. At 
each corner stands a pole, the height of the first being 30 

ellwood's test prob. — 10. 




146 TABLE BOOK AND TEST PROBLEMS. 

feet, the second 40, and the third 50. At what distance from 
the foot of each pole must a fourth pole be placed so that its 
top may just reach the top of each of the others ? How long 
is the fourth pole, the triangle being a horizontal plane ? 

313. A circle is inscribed in a quadrant. Required tlie 
radius of the next greatest circle that can be drawn in the 
quadrant, which shall be tangent exteriorly to this inscribed 
circle. 

GEOMETRICAL PROBLEMS, ETC. 

314. Construct the square root of any number JV". 

315. Construct the square root of 45. 

316. Construct the square root of 19. 

317. Let a square be inscribed in a circle, a circle in this 
square, a square in this circle, and so on. In such a construc- 
tion find the ratio of the diameter of the first circle to that of 
the seventh. 

318. Find the sum of the infinite series of diameters in 
the above. 

319. Demonstrate that the square inscribed in a semicircle 
is to the square inscribed in the entire circle as 2 to 5. 

320. Demonstrate that the square inscribed in a semicircle 
is to the square inscribed in a quadrant of the same circle as 
8 to 5. 

321. Demonstrate that the side of an equilateral triangle 
inscribed in a circle is to the radius as the V3 is to unity. 

322. Eequired the radius of the largest circle that can be 
inscribed in a triangle whose sides are respectively 11, 12, 
and 13 feet. 

323. Cut a board 10 feet long and 2 feet wide into four 
pieces which may be put together to form a square. 

324. A diameter of a circle is intersected by a chord. Show 
&at the square of the radius is equal to the square of the dis- 



MISCELLANEOUS PBOBLEMS. 147 

tance from the center of tlie circle to the point of intersection, 
plus the rectangle of the segments of the chord. 

325. Suppose a hemis]3herical vessel 3 feet in diameter were 
filled with water, and that ice would freeze uniformly 5 inches 
thick on the top and on the inner surface of the vessel. What 
would be the volume of ice thus formed ? 

326. If a circle be described touching the major axis of an 
ellipse in one focus, and passing through one extremity of the 
minor axis, half the major axis Avill be a mean proportional 
between half the minor axis and the diameter of this circle. 

327. The difference between the inscribed and circumscribed 
squares of a circle is 72 square feet. Find diameter of circle. 

328. Trisect the diagonal of a parallelogram. 

329. In the center of a rectangular lot 150 feet long and 80 
feet wide is a stake, to which a horse is tied by a rope 50 feet 
long. Over how much can the horse graze ? 

330. Three circular fields, each containing 80 acres, lie 
joining each other so as to inclose a small triangular piece 
of land. Mnd area of inclosed triangle. 

331. One acre of land lies between three equal circles drawn 
tangent to each other. Find the diameter of circles. 

332. Find the perimeter of a rhombus whose area is 216 
square inches, and one of whose diagonals is 24 inches. 

333. A ball 2 feet in diameter is in a corner. What is the 
diameter of the largest ball that can lie on the floor behind 
this one, both touching the same walls ? 

334. How high above the surface of the earth must a person 
be raised that he may see one third of the surface ? 

335. The three distances from a point within an equilateral 
triangle, to the angles, are a, h, and c. Determine the triangle. 

336. In a circle whose radius is 3, find the area of the part 
between parallel chords whose lengths are 4 and 5, both being 
on the same side of center. 



148 TABLE BOOK AND TEST PROBLEMS. 

337. By boring throngli the center of a sphere 4 inches in 
diameter, what volume will be bored away by an anger 2 inches 
in diameter ? 

338. Two circles touch each other, and also the base and arc 
of the semicircle in which they are inscribed. If their radii 
are a and h, what is the radius of the semicircle ? 

339. A hemispherical kettle of known and uniform thick- 
ness of shell is made from a given quantity of copper. Find 
an expression for the capacity of kettle. 

340. Demonstrate that the square described on the hypothe- 
nuse of a right triangle is equivalent to the sum of the squares 
on the other two sides. 

341. A conical glass 9 inches deep, and 6 inches wide at the 
top, is one third full of water. What is the radius of a ball 
that will just immerse ? 

TRIGONOMETRICAL PROBLEMS. 

342. At what latitude does a degree of longitude equal just 
half a degree at the equator ? 

343. What is the length of a chord cutting off one fifth of 
the area of a circle whose diameter is 10 feet ? 

344. Erom a point without a circular pond two tangents to 
its circumference are drawn, forming with each other an angle 
of 60°, a]^d the length of each tangent is 18 rods. Eind the 
diameter of the pond. 

345. A wheel 4 feet in diameter is sunk 1 foot in the mud. 
What portion of the area is in mud ? 

346. A ship, starting from the equator, sails due northeast 
1000 miles. Supposing the earth a sphere, find the ship's 
latitude. 

347. Eind the angle of elevation, and velocity of projection, 
of a shell, so that it may pass through two points : the coordi- 



MISCELLANEOUS PBOBLEMS. 149 

nates of the first being x' = 1700 feet, y' = 10 feet ; and of the 

second, x'' = 1800 feet, y" = 10 feet. (Olmsted's College Philosophy.) 

348. Upon how much surface can a horse graze when tied 
by a rope 100 feet long to a corner of a barn 25 feet square ? 

349. If a heavy sphere, whose diameter is 4 inches, be let 
fall into a conical glass full of water, whose diameter is 6 and 
altitude 6 inches, how much water will run over ? 

350. A horse is tied to a stake in the circumference of a 
circular 10-acre field. How long must the line be to allow him 
to graze over one acre inside the field ? 

351. Twenty acres are inclosed in circular form, and a 
stake is driven 10 feet from the circumference. How long a 
rope fastened to the stake will allow an animal to graze on 
one acre inside the fence ? 

PROBLEMS INVOLVING CALCULUS. 

352. Find the number whose nth root exceeds itself by the 
greatest possible quantity. 

353. The side of a square inscribed in a right triangle 
is 12.' What are the sides, if the hypothenuse is the least 
possible ? 

354. How far above the center of a horizontal circle whose 
radius is 2^2 rods, must a light be placed that the illumina- 
tion may be a maximum ? 

PROMISCUOUS PROBLEMS. 

355. Show that 9.45 = 9^^. 

356. A flash of lightning was seen 8 seconds before the 
thunder was heard. How far away was the cloud ? 

357. How far will a nail head in the tire of a wheel move 
in driving 3.1416 rods ? 

358. A can dig a row of potatoes while B weeds 1. B can 
dig a row while A weeds 4. If both receive f 6 a day, what is 
the share of each ? 



150 TABLE BOOK AND TEST PBOBLEMS, 

359. A composition of gold and silver which weighs a 
pounds loses m pounds in water. ISTow, a pounds of gold lose 
n pounds in water, and a pounds of silver lose p pounds in 
water. How many pounds of gold in the composition ? 

360. A and B can dig a ditch in 12 days ; B and C, in 20 
days ; and A and C, in 15 days. How long would it take all 
of them to dig it ? How long each ? 

361. Three poles, each 50 feet long, were erected on a plain 
so that the upper ends met, and the lower ends were 60 feet 
apart. What length of plumb-line was required to reach from 
their point of meeting to the ground ? 

362. Bought $7000 worth of bonds due in 20 years, the 
interest on which was 7 per cent, payable semi-annually. They 
yielded me 8 per cent, payable semi-annually. What did I 
pay for them ? 

363. Given x'-6x=7, 
to find X without completing the square. 

364. Given x^ + 2x-8 = 0, 
to find X without completing the square. 

365. How much longer is the fence around a rectangular 
10-acre field, whose breadth is one fourth the length, than 
the fence around a square field containing the same number of 
acres ? 

366. Satisfy the conditions in the equation 

x^ + x = £i square number. 

367. A man jjulls out the stumps in a field at the rate of 
25 cents apiece, and piles the stones at 10 cents a score. One 
stump occupies him as long as 40 stones. He works 3 days 
and earns $ 8, then goes on at the same rate of working, and 
finishes the job in 3| days more, earning altogether $ 20. How 
many stumps and stones were there in the field? 



PBOBLEMS WITH CURIOUS RESULTS, 151 



PROBLEMS WITH CURIOUS RESULTS. 

DIGITS. 

368. Multiply the number 12,345,679 by some number that 
will make the product contain but one of the nine digits. 

369. Arrange the nine digits so that their cube root can be 
extracted. 

"ONE CENT." 

370. Suppose one cent to have been placed at 6% compound 
interest at the commencement of the Christian era. What 
would the amount have been Jan. 1, 1882 ? 

(From the late E. B. Seitz, in Mathematical Magazine.') 

INVOLUTION OF IMAGINARY QUANTITIES. 

371. Show the absurdity of assuming that V— a^ = a;. 



372. rind the values of (V-1)^ (V-1)^ (V--l)^ 

373. Why does minus multiplied by minus give plus ? 

THE ZERO FACTOR. 

374. Prove the absurdity of the equation a = x, 

375. Prove the absurdity of the proportion —a:a::a:—a, 
in which a is any finite quantity. 

SOMETHING TO INVESTIGATE. 

376. It is well known that any number of figures, multiplied 
by 3 or any of its multiples, and the digits of the product 
added until a single digit results, will give as a result either 
3, 6, or 9. Exemplify this. 



162 



TABLE BOOK AND TEST PROBLEMS, 



THE PROPOSITION OF ARCHIMEDES. 

377. "Archimedes, the celebrated philosopher and mechanic 
of Syracuse, once exclaimed, ' Give me a place to stand, and I 
will move the world.' '^ Prove his inability to do this. 

(From Professor B. F. Burleson of Oneida Castle, N.Y., in Notes and Queries.) 



1888. 

378. Substitute numbers for these 
letters, so that when added in any one 
of the following ways the sum will be 

1888 : — 



a + 5 + c + d. 
d +li + l +p. 

6 +f+g + ^^^ 

a +/+A:+i?. 
c -\-g + k + o, 
k +1 +o+p. 
a +6 + +_p. 
m + i +d + h, 
m + i +k + 0. 



a + e + i + m. 

p + +n + m. 

i +j +Jc +L 

(^ + 9 +j +^- 
a + b+e +/. 

i + J + m + n. 
c + d + wi + n. 
a + G + e +g. 
d + h + h +/. 



a 


b 


c d 


e 


f 


9 


h 


^ J 


k I 


m 


n 





P 



^ + f+j+n. 
c + d + g-\-h. 

f+g+j +T^- 
p + l +e + a, 
p + l J^n + g. 



SUMMATION BY SUBTRACTION. 
379. Find the sum of 379, 8452, 31, and 60 by subtraction. 



SEBIES. 153 



SERIES. 

380. Sum the series — = 1 = 1 1 etc. 

2. 3. 44. 5; 66. 7. 8 

381. Find the sum s of the series 

1 + 2 + 3 + 4 2+3+4+5 n+(n + l) + (n+2) + (n + 3) 

1.2.3.4 2.3.4.5 "^ n{n + l){n+2){n + 3) 

382. Find four numbers in proportion such that their sum 
shall be a, the sum of their squares b, and the sum of their 
cubes c. 

383. The sum of the terms in an arithmetical series is 
s = 1146, the sum of their squares is b = 126,746, and the sum 
of their cubes is c = 15,409,116. Find the first term a, the 
common difference d, and the number of terms n. 

384. The sum of the terms in a geometrical series is 5 = 
2059, the sum of their squares is 6 = 953,317, and the sum of 
their cubes is c = 550,434,529. Find the first term a, the 
ratio r, and number of terms n, 

385. The sum of n terms of the series (2^ . 4^ - 1^ . 3^) + 
(62.82-52.72) + (102.122-92.112)+... etc., is s = 22,395,834,- 
549,559. Find the number of terms w. 

386. Find the 7ith. term, and the sum of 71 terms, of the 
series 1, 19, 10, 14|-, l2i, etc., in which each term after the 
second is an arithmetical mean between the preceding two. 

387. Find the nth term, and the continued product of n 
terms, of the series 4, 64, 16, 32, 16V2, etc., in which each 
term after the second is a geometrical mean between the pre- 
ceding two. 

388. Given ix — x'^ ^1 ^^ g^^^ ^ series the 

l + 2a;-3a;2 + 2a;^ a 
value of X when a is greater than 1. 



154 TABLE BOOK AND TEST PBOBLEMS. 

389. Sum to n terms the series whose general term is 
n^(3n — 2)x''~^, and find the numerical value of the same when 
x = 5, and n= all integral values from 1 to 100; also when 
X = .999, and n = all integral values from 1 to oo. 

390. A note of $ 400 at annual interest amounted to $ 441.60 
in 4 years. Required the rate. 

391. A person who enjoyed a perpetuity of $1000 per 
annum provided in his will that after his death it should 
descend to his son for 10 years^ to his daughter for the next 
20 years, and to a hospital forever afterwards. What was the 
value of each bequest at the time of his decease, allowing com- 
pound interest at 6 per cent ? 

392. The number of balls in the rth = 13th course of a com- 
plete rectangular pile of cannon balls is & = 32, and the 
number in the qih = 7th course is c = 140. How many balls 
in the pile ? 

393. If the first pair of doves had produced not any at the 
end of their first year, 2 pairs at the end of their second, 3 
pairs at the end of their third, not any at the end of their 
fourth, 2 pairs at the end of their fifth, 3 pairs at the end of 
their sixth, and so on, repeating the numbers 0, 2, 3 in regular 
order ; and if each pair of doves produced had bred in a simi- 
lar manner, and none had died, — how many pairs of doves 
would there have been at the end of the first century ? 

1 2 _i Q2 92 _|_ /t2 *^2 i_ K2 

394. Sum the series :?-±^ + =-±^ + ^-^^ + etc. 



Part III. 

SOLUTIONS 



Note. — The solutions are numbered to correspond with the problems in Part II., 

to which they refer. 

ARITHMETICAL SOLUTIONS. 
DENOMINATE NUMBERS. 

1. 3 reams = 1440 sheets ; 5 quires = 120 sheets ; the total = 1579 
sheets. Since there are 20 quires in a ream, and 1 quire to a ream is 
wasted, the waste = 2V of 1579 = 79-. 1579 - 79 = 1500, the number 
used. Each sheet makes 2 posters : hence 1500 x 2 = 3000, the number 
printed. 

2. For every pound he sells 15J ounces. 232 pounds or 3712 ounces 
-f- 15.75 = 235|f , the number of fraudulent pounds he sells. 235|f - 232 
= 3|| fraudulent pounds. 18 1 x 3|f = 692t cents, the gain. 

3. The Troy pound, by which gold is weighed, contains 5760 grains; 
and the Avoirdupois pound, which is used in weighing lead, contains 7000 
grains : hence the lead pound is the heavier. 

7000 grains -^ 16 = 437J grains = 1 ounce Avoirdupois. 
5760 grains -4- 12 = 480 grains = 1 ounce Troy. 
Hence the ounce of gold is the. heavier. 

4. As his watch is too slow by 6| hours, he must be 15 x 6J = 97^30' 
east of Philadelphia, or 97° 30' - 75° 10' = 22° 20' east longitude. He 
must therefore have travelled through 360° - 97° 30' = 262° 30'. 

5. Contents of cube = 2x2x2 = 8 cubic feet. 1000 x 8 = 8000, 
weight of same bulk of water. 8000 x 19.36 = 154,880 ounces = 9680 
pounds. 

Notes. — 1. The specific gravity of any substance is its weight compared with the 
weight of an equal bulk of water. 

2. Assume that a cubic foot of water weighs 1000 ounces; and a cubic foot of air at the 
earth's surface, about 1.22 ounces. 

1.^5 



156 TABLE BOOK AND TEST PROBLEMS, 



LEAST COMMON MULTIPLE AND GREATEST COMMON 

DIVISOR. 

6. L. C. M. of 3, 5, and 7 = 105 ; G. C. D. of 4, 6, and 10 = 2 : 

.-. 105 - 2 = 52}. Ans. 

7. G.C.D. of 2, 4, and 8 = 2 ; L. CM. of 3, 7, and 9 = 63 : 

.-. the required G. C. D. = 2 -^ 63 = ^j. 

8. The L. C. M. of 3, 7, 12, and 14 = 84 : hence they will toll together 
once in every 84 minutes ; and in 7 hours, or 420 minutes, they will toll 
together 5 times, not counting the first tolling. 

9. A makes a trip every 4 days, B every 6 days, and C and D every 8 
days. The L. C. M. of 4, 6, and 8 = 24 ; that is, they start out together 
every 24 days. But this includes the stay-at-home day ; hence 24 — 1 = 23, 
the required number of days. 

10. B will overtake A once in every 320 -^ 5 = 64 minutes. A and C 
will meet once in every 320 -^ (8 + 24) = 10 minutes. Therefore all will be 
together in the L. C. M. of 64 and 10, which is 320 minutes, or 5 hours 
20 minutes. 

PARTNERSHIP. 

11. Being equal partners, they must share all indebtedness equally: 
hence A should give to B J of $240 + } of $260, or $250. 

12. Had A owned J of the stock, his gain would have been f of $ 249 
r= $186.75. $186.75 - $175 = $11.75 = the gain on $10 of the stock: 
hence $1 of stock gains $1,175. To gain $249 requires $249^1.175 
= $211.91, the whole stock. 

13. This was a partnership, A owning ■^■^, B J, and C \\. They have 
for division 3 loaves and 24 cents : hence A should have f of a loaf and 5 
cents ; B, 1 loaf and 8 cents ; C, y- of a loaf and 11 cents. But each ate 
1 loaf: hence B should get 8 cents. A, having eaten f of a loaf more 
than his share, must deduct the price thereof from his 5 cents. Since 
each loaf cost 6 cents, f of a loaf is worth f of 6, or 2} cents; and 5 — 2} 
= 2J cents = A's share. The 2} cents go to C, who supplied the bread to 
A : hence C's share = 11 + 2} = 13] cents. 

14. First Solution. If Smith had been a partner all the way, he would 
have paid \ of $ 10, or $ 5. But since he rides only half the distance, he 
should pay only J of $5, or $2 J. 



ARITHMETICAL SOLUTIONS, 157 

Second Solution, Jones rides 20 miles, and Smith rides 10 miles. 
Therefore, since Jones rides twice as far as Smith, he should pay twice 
as much. Hence Jones should pay $6|, and Smith $3|. 

PROPORTION. 

15. If J of 6 were 3, f of 6 would be 9, and J of 6 would be 4|. If 3 
were J of 6, 9 would be f of 6, or 6 ; and 6 would therefore be 4 : hence 
^ of 6 would be 2. The two statements are different. In the first case 
the condition increases our numbers 50 per cent, while in the second 
instance it decreases them 33 J per cent. The problem might be written 
thus : (a) If 2 were 3, what would 3 be ? (6) If 3 were 2, what would 
3 be ? By proportion we have, 

2:S::S:x = 4:i,Ans.to (a); 
and 3:2::3:x = 2, Ans. to (b). 

16. Suppose n = number of men that carry off the influx. As the 
number of men increases, the time of emptying decreases. Hence we have 
this proportion : 12 - n:16 - n: : SO: 40. .-. n = S, Then 12 - 3 (= 9) 
men empty the cistern (without the influx) in 40 minutes, or 1 man does 
it in 360 minutes. Hence 18 — 3 (= 15) men can empty it in 360 -f- 15 = 24 
minutes. 

17. In all such problems the weights borne are inversely as the dis- 
tances of the carrying points from the point of suspension. Hence, A 
being IJ feet, and B 2 J feet, from the point of suspension, A carries 5 
pounds to B's 3. Therefore A carries 125 pounds, and B 75 pounds. 

18. Prom the earth's center to the given height is IJ times the radius 
of the earth. Since gravitation varies inversely as the square of the dis- 
tance between the centers of gravity, we have the proportion 

(1 J)2 : 12 : :1800 : x, the required weight. 
Whence x = 1800 -r- f = 800 pounds. 

19. Since A wins 9 while B wins 3, B's skill is J of A's ; and since A 
wins 5 to C's 3, C's skill is | of A's. Therefore B's skill is to C's as J to 
f , or 5 to 9. Hence, out of 70 games, B should win 25, and C 45. 

20. 20 : true weight : : true weight : 31 J ; whence the square of the true 
weight = 20 X 31i = 625, and the true weight = V625 = 25 pounds. 

21. The sectional area being unchanged, the volume and velocity of 
air currents are inversely proportional to the square roots of the lengths. 



158 TABLE BOOK AND TEST PBOBLEMS. 

Hence V4000 : VIOOO : : 20,000 : (x), 

or 2: 1:: 20,000: (x); 

whence x = 10,000 cu. ft. 

22. We assume the clock to be on the equator. The third law of the 
pendulum is this : The lengths are inversely proportional to the squares 
of the numbers of vibrations in a given time. The proportion is ex- 
pressed thus: L:l:: n^: iV'^. The length of a second's pendulum at the 
equator is 39 inches : hence we have the proportion 15 feet (=180 inches) : 
39 inches : : 1 : x, whence x — -^^^ ; that is, the 15-foot pendulum vibrates 
y\^o of a time in 1 second. In a minute it vibrates -^^q x 60 = 13 times. 
Since it loses 3 hours, or 180 minutes, in a week, it loses 13 x 180 = 2340 
vibrations. In 1 hour it loses 2340 -r- 168 = 13.928 vibrations. It now 
vibrates 13 x 60 = 780 times in an hour, and to insure correct time it 
must vibrate 780 + 13.928 — 793.928 times. Hence this proportion: 
15: Z: : (793.928)2; (780)2, whence I = 14.469 = length of pendulum after 
correction. Hence the "bob" must be raised 15 — 14.469 = .531 of a 
foot = 6.372 inches. 

PROFIT AND LOSS. 

23. The present worth of the note is $240 -4- 1.20 = $200. Hence A 
gains $200 — $190 = $10, and interest on the same for 4 years at 5 per 
cent, which is $2. 

24. Let 100 per cent = cost. Then | — J = f of selling price = cost = 
100 per cent. J = | of 100 per cent = 14f , and | = 114f per cent = selling 
price. Then 114f — 100 = 14f per cent = rate of gain. 

25. 180 per cent of first = 120 per cent of second = 90 per cent of 
third. Hence the third cost twice the first, and the second 1 J times the 
first. Then the 1st + twice the 1st + IJ times the 1st = 4J times the 
1st = $90, and the 1st = $20. $20 x 2 = $40 = 3d. $20 x 1J = $30 = 2d. 

26. Fii'st Solution. By the problem, f of cost = selling price. By 
conditions, the selling price is 115 per cent of real cost, plus 115 per cent 
of $500 ; or |J of cost + $575 = selling price. Therefore $575 is the dif- 
ference between § of cost and |^ of cost, or -^^ of cost. Hence the cost 
was $575x20 = $11,500. 

Second Solution. Difference in rate of gain = 20 per cent — 15 per 
cent = 5 per cent, which is $500 + 15 per cent of $500, or $575. Then 
1 per cent = ^ of $ 575 = $ 115, and 100 per cent = $ 11,500. 



ARITHMETICAL SOLUTIONS. 159 

27. 100 per cent less 10 per cent = 90 per cent. 

90 per cent less 10 per cent = 81 per cent. 
81 per cent less 5 per cent = 76.95 per cent. 

150 per cent of 76.95 = 115.425 per cent of the original cost. 

28. (^ 37.50 X 30) + ($21.60 x 20) = $1557 = selling price. $ 1557 - 
$ 1500 = $57 = gain = 1 ox and 1 cow. Then $57 x 20 = $1140 = 20 
oxen and 20 cows. Hence 10 oxen must have cost $ 1500 - $1140 = $360, 
and 1 ox cost $36. Then $57 - $36 = $21, cost of 1 cow. 

29. 120 per cent of real sales = 140 per cent, of supposed sales. 

1 per cent of real sales = 1 J per cent of supposed sales. 

100 per cent of real sales = 116| per cent of supposed sales. 

Hence he sold 16 1 per cent = i more than he intended to sell, and the 
length of his yard-stick was 36 x 1^ = 42 inches. 

30. $16 - $10 = $6 = IJ times B's gain. Hence B gains, $4, and the 
article cost him $12. His rate of gain is y\, or 33 J per cent. Then 
$ 12 = 133 J per cent of what A paid. Hence A paid $ 12 -^ 1.33^ = $9. 

31. $1 + $.50 = $1.50 = gain by the 3-cent increase. An increase of 
1 cent per pound would give a gain of $1.50 -^ 3 = 50 cents. Hence 
50 -T- 1 = 50 = the number of pounds. 

32. Let 100 per cent = the boat. Then the tannery = 116 per cent, and 
the brewery = 116 per cent x .96 = 111.36 per cent. 75 per cent of the 
brewery = 83.52 per cent of the boat. 100 per cent — 83.52 per cent 
= 16.48 per cent, the loss. .-. 16.48 per cent of the boat = $103, 1 per 
cent = $6.25, and 100 per cent = $625. Tannery = $625 x 1.16 = $725. 

33. Marked price = 150 per cent = 120 per cent of the cost of real 
amount sold. 150 -^ 120 = 1.25 = 1 per cent ; and 125 per cent = 100 per 
cent, or what is sold for a yard. Hence 125 — 100 = 25 per cent, the 
amount sold by mistake, =9 inches. Then 36 + 9 = 45 inches, length of 
yard-stick. 

34. Let 100 per cent = value of goods. Then 80 per cent = cost. 50 per 
cent of 80 per cent = 40 per cent of value = the entire gain. Hence 40 
per cent — 20 per cent = 20 per cent = the "honest " gain. 

35. Let 100 per cent = original value. Then 70 per cent = cost, and 
52.5 per cent = selling price. 100 per cent — 52.5 per cent = 47.5 per cent 

of first value = $55. Hence original value was $55 x = $115^f. 

($115J-f X .70) X .75 = $60.78if. ^^'^ 



160 TABLE BOOK AND TEST PBOBLEMS, 

36. Let 100 per cent = the cost. Then 114 1 per cent = the selling price. 
For 14 pounds he pays the price of 13, or || of 100 per cent = 92| per 
cent. He sells 13 pounds for the price of 14, or if of 114| per cent =123ji3 
per cent. The gain is 123^13 — 92f = 30 |f per cent. The gain ought to 
have been 14 f^ per cent. Therefore 30|J — 14f^ = 15 |f per cent = gain by- 
fraud = $29 ; and 100 per cent = $182, the required cost. 

37. First Solution. Let 100 per cent = the cost. Then 80 per cent = 
supposed cost. Now, 30 per cent of 80 per cent = 24 per cent of 100 per 
cent. Hence, if the 20 per cent (supposed decrease) yield 24 per cent, 
100 per cent yields 120 per cent. Therefore 120 per cent — 100 per 
cent = 20 per cent = gain. 

Second Solution. gV of selling price — -^^-^ of same = ^Jq of selling 
price = difference in rates = 30 per cent = j^^^. Whence selling price = 
120 per cent of cost. .-. gain r= 20 per cent. 

38. First Solution. $200 - $100 = $100 = part of gain. 3 per cent 
of $100 = $3, which is also gain, being interest. Hence $103 is the gain. 

Second Solution. He gains $100 + the interest of $200 for 6 months 
at 6 per cent, or $ 106. 

Third Solution. Must A pay interest for 6 months on $100 ? If not, 
the following is correct: $100 ^ 1.03 == $97,087, the present worth. 
$200 - $97,087 = $102,913, the gain. 

Fourth Solution. Let A set apart $ 100 of his cash to meet his pay- 
ment at the end of 6 months. The other $ 100 is his gain, and it is his 
only gain. 

39. If I get interest from purchaser, I receive $ 103 in 6 months, which 
is equivalent to $100 now. Hence $200 - $100 = $100, the loss. If I 
do not get interest, I receive $ 100 in 6 months, the present worth of which 
is $97,087. Hence I lose $200 - $97,087 = $102,913. 



STOCKS AND BONDS. 

40. Each bond yields $5 interest yearly. If $5 = 7 per cent of cost, 
100 per cent = ^^^ of $5 = $71f. $100 - $71f = $28^-, the discount. 
$28^ -T- 100 = 28 f per cent, the rate of discount. 

41. 5 per cent semi-annually = 10 per cent yearly. Then $2000 = in- 
come = 10 per cent of the face value of the stock, and $20,000 = face = 
2000 shares. 1 share costs $10 x 1.20} = $12.0|, and 2000 shares cost 
12.0} X 2000 = $24,050. 



ARITHMETICAL SOLUTION d. 161 

42. $21,200 -^ 106 = 200 shares. Each share yields $6 in gold yearly. 
Hence $6 x 200 = $1200, income in gold. $1200 x IJ = $1350, income 
in currency. $6 in gold = $6.75 in currency ; and $6.75 -f- 106 = 6j%\ 
per cent. 

43. Each IT. S. 5 brings in a yearly interest of $5. But since the cost 
of the bond was 75 per cent, or $75 for a $100 bond, the $5 is the yearly 
interest of $75. Hence 5 -f- 75 = y^^- = 6f per cent = rate of income. 
Each IT. S. 6 produces $6 yearly. But when bought at 85 per cent, the 
rate of income is 6 -=- 85 = 7j\ per cent. Hence the 6's are 7^j per cent 

— 6 1 per cent = |f of one per cent, the better investment. 

44. The interest on each bond for 15 years = $75, and the loss by 
selling at par = $10. Therefore the net gain is $65 in 15 years, or $4J 
in 1 year. If the investment, $110, yields $4 J, the rate is 4] -^ 110 = 
3|J per cent. 

45. 117f X 2000 = $2352.50 = cost of first kind of bonds. Then $5000 

- $2352.50 = $2647.50 = remainder. $2647.50 -- llOJ = 23J bonds, and 
a surplus of $50.75. The interest on the $2000 6's = $120 annually. 
The interest on 23 J bonds at 4} per cent = $105.75. Then $120 + 
$105.75 = $225.75 = annual income. 

46. Cost of first kind of stock = 107 J + J = 108 : hence the gain on $ 1 
= j^j = J^ of $1. Cost of second kind of stock = 98^ + J = 99 : hence 
gain on $1 = -^g of $1, and on $2 it is i§. Then the gain on every $3 
invested is ^^ + i§ = y^g- of $1. Consequently $3 were invested as often 
as jW is contained in $3348, or 21,384 times ; that is, the whole amount 
invested was $64,152, of which I, or $21,384, was invested in the first 
kind of stock, and f, or $42,768, in the other. 

INTEREST. 

47. A's fortune = f of B's. f of A's = f of B's. Then B's + f B's = 
y- of B's == amount on interest. Amount of $ 1 for 6 years at 8 per cent 
= $1.48. $8880 -- 1.48 = $6000 = amount on interest. Hence -\^- of B's 
money = $6000, and B's = $4153}!. | of $4153}i = $2769y% = A's. 

48. It is evident that the note must be drawn for $ 60 + the interest. 
It is also evident that 6 per cent of $60 + 6 per cent of the interest must 
be the interest. Hence $ 3.60 + 6 per cent of the interest = the inter- 
est, or $3.60 = 94 per cent of the interest. The interest, therefore, is 
100(3.60^94) =$3,829, and the face of the note is $60 + $3,829 = 
$63,829. 

11. 



162 TABLE BOOK AND TEST PBOBLEMS, 

49. The interest for 2 years at 5 per cent — J^ of the principal. Since 
the $ 730 includes -^^ of the note, and also part paid of the debt, ^ 1000 
— $730 = $270 = j^o of note. Hence the note must be given for $270 
X ^-==$300. 

50. In two years the holder of the estate receives two payments of 
$ 100 each, one of which might have been put at interest for 1 year at 5 
per cent. Therefore at the end of the second year he receives $205 ; and, 
letting S = the sum to be paid out biennially, $ 205 — S = the net income, 
which is equal to the compound interest of $1000 for 2 years, or $102.50. 
Hence $ 205 - /S' == $ 102. 50, or ;S' = $ 102. 50. 

51. Interest of $1 for 93 days at 6 per cent = .0155. Proceeds of $1 
= $ 1 - .0155 = .9845. Hence in 93 days the bank makes .0155 on .9845, 
which is equivalent to about 6.2 per cent. 

Note. — By the interest tables, the interest on $1 for 93 days at 6 per cent is 2 cents. 
Then the bank makes 2 cents on 98, or about 8 per cent. 

52. The compound amount of $ 1 for 3 years at 6 per cent is $ 1.191016, 
and for $6000 it is $1.191016 x 6000 = $7146.096. Then $1 + $1.06 + 
$1.1236 = $3.1836 ; and the annual payment = $7146.096 -- $3.1836 = 
$2244.66. 

53. Let 100 per cent = the principal. The difference in time is 1|- 
years. Interest on 100 per cent for IJ years at 8 per cent = 10 per cent. 
Then $ 50 — 10 per cent = interest on principal at 2 per cent for the 
required time, or 

$25 — 5 per cent = same at 1 per cent (1) 

1 year's interest = 6 per cent, and the entire interest is $325 — 100 per 
cent, which divided by 1 year's interest gives the time from date of note 

to May 1, 1885. Hence ^^'^^ - 100 per cent ^ ^^^^ number of years. At 

6 per cent 
1 per cent the interest on the principal for 1 year is 1 per cent, and for 
the required time it is 

$325 -100 per cent ^ ^ ^^^^ ^ $3.25 -1 percent .g) 

6 per cent 6 per cent 

Now, equating (1) and (2), we have ^-"^-25-1 per cent ^^^^ _ ^ ^^^ 

6 per cent 
cent. Multiplying both sides by 6 per cent, or jJ^q, we have $3.25 — 1 
per cent = $1.50 — .3 per cent. Hence $3.25 — $1.50 must be 1 per 
cent — .3 per cent, or .7 per cent = $ 1.75. Then 1 per cent = $1.75 x Y- 
= $2.50, and 100 per cent = $250, the face of the note. The interest for 



ARITHMETICAL SOLUTIONS, 163 

1 year = ^15: hence to produce $325 - $250 = $75 requires 75 -r- 15 
= 5 years. The date must therefore be 5 years prior to May 1, 1885, or 
May 1, 1880. 

DISCOUNT AND PRESENT WORTH. 

54. First Solution. By bank discount, the interest of $1000 for 63 
days := $ 10.50. Hence the present worth = $ 1000 - $ 10.50 = $989.50. 

Second Solution. By true discount, the amount of $ 1 for 60 days = 
$ 1.01. Then $ 1000 - $ 1.01 = $ 990.099+ = present worth. 

55. Amount of $150 for 9 months at 8 per cent = $159; amount of 
$ 200 for 1 year 9 months at 8 per cent = $ 228 ; amount of $ 200 for 2 
years 9 months at 8 per cent = $244. The present worth of $159 at 6 
per cent = $152. 1 5 + ; the present worth of $ 228 at 6 per cent = $ 206.33 + ; 
the present worth of $244 at 6 per cent = $209.44 + . Hence the cash 
value was $152,153 + $206,334 + $209,442 + $50 = $617,929. 

56. 120 per cent of $ 500 = $ 600, the money I received. Discount on 
$1 for 183 days = .0305. $1 - .0305 = .9695, proceeds of $1. $600 ^ 
.9695 = $618if|f, face of note. 

57. Amount of $1 for 63 days = $1.0105. .06 -- 1.0105 = 5.937+ per 
cent. 

58. Interest of $1 for 33 days at 6 per cent = .0055. $ 1 - .0055 = .9945, 
the proceeds of $ 1. Hence $2000 -^ .9945 = $2011.06, the face. 

Note. — Many banks ignore fractions of cents, and take 1 cent as the interest of $1 for 
33 days. Using this interest instead of $.0055, we find the face of the note to be $2020.20. 
Some banks count also the day of discount, which makes 34 days. 

59. In bank discount the face of the note is equal to the proceeds plus 
the interest on the proceeds for the given time at the rate received at the 
bank. Hence the face value of $ 1 of proceeds for 33 days at 6 per cent is 
$1.0055. Now, if $1 yield a certain sum at 6 per cent, $1.0055 must 
yield the same amount in the same time at .06 -f- 1.0055 = 5i|ff per cent. 

60. Reversing the process in the preceding solution, we find the face 
value of $1 of proceeds to be .065 -^ .06 = $1.083|. Hence the interest 
of $ 1 for the required time at 6} per cent = $ 1.08 V - $ 1 = .08 J ; for 1 
year it is .065, hence the required time is .08 J -r- .065 = 1.28205+ years 
= 1 year 3 months 12 days. 

61. The shares were at interest 8, 6, and 4 years. The compound 
amount of $1 for 8 years at 6 per cent is $1.593848, and the present 
worth of $1 for 8 years is $1 -- 1.593848 = .62741+ ; for 6 years it is 
$1 -T- 1.418519, or .70496 ; for 4 years it is $1 - 1.262477, or .79209. It 
is easily seen now that the shares must be to each other in the proportion 



164 



TABLE BOOK AND TEST PBOBLEMS. 



of these present worths. Hence A, the youngest, gets /f^aWe' ^ AWA* 
and C aWA'c* of 19000. A, therefore, gets ^^ 2657.94+ ; B, $2986.47+ ; 
and C, $3355.59-. 

INVOLUTION AND EVOLUTION. 

62. When we point off decimals, we must count from the decimal 
point. Hence \/.'625 = .79 + . 



63. I of a number cubed 



^r 



of the cube of the number. Then § of 



the cube, or |f, — 27 = 10, and if = 10j_ whence |f 
number. The number, therefore, is \/27 = 3. 



27, the cube of the 



64. The square of | of a number = j% of the square of the number, 
and I of this = | of the square of the number. The square of J a number 
= J of the square of the number, and J of this = j\ of the square of the 
number. By the problem, f of the square of the number is 12 more than 



-^\ of the same. Hence f — y^^ = -^^ of the square of the number = 12, 



and i| = 12 X y = 64, the square of the number. Then V64 = 8 = the 
required number. 

65. 402 ^ 1600 ; i of 1600 = 400 = area of walk. 

1600 - 400 rrr 1200 = area inside of the walk. 
\/l200 = 34.64 = length of one side ; 
then 40 - 34.64 = 5.36 = twice the width of walk. 

Hence 5.36 -~2= 2.68+ = width of walk. 



66. 



First Solution. 

(a) (b) 



16 



ri3 

^18 
121 



i 




2 






i 




5 


i 




3 






i 




2 






1 


7 



Second Solution. 

(a) {b) 



10 i 



13 


1 

3 




5 




14 




I 




1 


18 




i 




1 


.21 


i 




3 

8 


2" 



ALLIGATION. 

5x3 + 7x5 = 50: hence multiply column (a) 
by 3, and column (b) by 5, which gives 

6 at 13 = .78 

25 at 14 = 3.50 

9 at 18 = 1.62 

10 at 21 =2.10 

50 $8.00 

Multiplying both columns by 5, we have 

25 at 13 

5 at 14 

5 at 18 

15 at 21 



ARITHMETICAL SOLUTIONS. 165 

Or multiplying column (a) by 2, and column (b) by 17, gives 

10 at 13 
17 at 14 

17 at 18 
6 at 21 

Or multiplying column (a) by 6, and column (b) by 1, gives 

30 at 13 
1 at 14 
1 at 18 . 

18 at 21 

Or multiplying column (a) by 1, and column (6) by 21, gives 

5 at 13 

21 at 14 

21 at 18 

3 at 21 

Or multiplying column (a) by 3, and column (5) by 13, gives 

15 at 13 

13 at 14 

13 at 18 

9 at 21 

67. Since 1 cubic centimeter of water weighs 1 gram, the specific gravity 
of any substance is the weight of a cubic centimeter in grams. Then 1 
cubic centimeter of gold weighs 19.25 grams ; of silver, 10.5 grams ; and 
of the alloy, 16.84 grams. By alligation 

6.34 volumes of gold. 
2.41 volumes of silver. 



ri9.^ 

(lO.^ 



19.25 
16.84 ■ 

.50 



Now, 19.25 X 6.34 = 122.045 grams of gold, 

and 10.5 x 2.41 = 25.305 grams of silver : 

hence the volumes of the relative quantities of gold and silver are as 634 
to 241 ; and the weights as 122,045 to 25,305, or as 3487 to 723. 

(By Dr. I. J. Wireback.) 



166 TABLE BOOK AND TEST PROBLEMS. 



ANNUITIES. 

68. $300 -T- 12 = $25.00, the monthly rent. 

12 X 13 

— X 25 = $ 9.75, the interest on payments. 

$300 + $9.75 = $309.75, the finaUvalue. 
$309.75 ---1.06 = $292.22. 

69. To give an income of $ 1200 a year requires $ 1200 h- .06 = $20,000. 
The compound interest of $ 1 for 6 years at 6 per cent is $0.418519, which 
multiplied by 20,000 gives $8370.38, the final value. $8370.38-- 1.418519, 
the compound amount of $1 for the given rate and time, = $5900.79, the 
present value. Hence $6000 - $5900.79 = $99.21, the amount B gains 
by installment plan. 

70. The annual payment must be an annuity, the amount of which in 
6 years will be 

1000 X (1.06)5 (1) 

If no payments were made, there would be due at the end of 5 years the 
compound amount of $ 1000, or 

1000 X (1.06)5 (2) 

This amount is composed of five amounts, as follows: 1. The first pay- 
ment at compound interest for 4 years; 2. The second for 3 years; 
3. The third for 2 years ; 4. The fourth for 1 year ; 5. The fifth payment. 
The amount of an annuity of $ 1 for 5 years is 

(1.06)4 _|_ (1.06)3 + (1.06)2 + (1.06)1 -f (1.06)^ 

The sum of this geometrical progression is 

(L06)^-2 (3) 

.06 ^ "^ 

If $ 1 amounts to (3) , it will require as many dollars to amount to (2) as 
(3) is contained times in (2), or 

1000 X (1.06)5 ^ 1000 X .06 X (1.06)5 ^ 
(1.06)^ - 1 (1.06)5 _ 1 ^ . . 

.06 

"AGE" PROBLEMS. 

71. J of yours -f 20 = J (yours + 20), or \ of yours + 20 = ^ yours -f 
10 ; whence J of yours = 10, yours = 40, and mine = } of 40 = 10 years. 



ABITHMETICAL SOLUTIONS, 167 

72. Ten years ago twice her age was the difference between their ages, 
and her age was J the difference. Now his age is 2i times hers, and IJ- 
times her age is the difference, or her age is f the difference. As the 
difference always remains the same, 10 years must be f -^ J == J of the 
difference : hence the difference is 40 years, which is twice her age 10 
years ago. 

73. First Solution. Twice Edwin's age = the difference between their 
ages 10 years ago, and once his age = J the difference. But 12 years 
later Edwin's age equals the difference. Hence 12 years = J the differ- 
ence ; and the difference is 24 years, which is twice Edwin's age 10 years 
ago. Edwin is now 12 -f- 10 = 22 years of age. 

Second Solution. Let F and F represent their ages 10 years ago. 
Thenii^-£', andii^+ 12 = J (^ + 12), orf -P+24 = F -\- 12. Hence 
|i^ = 12, Edwin's age 10 years ago, 

74. This problem is found in Kaub's *' Complete Arithmetic," p. 323, 
and is generally regarded as hisolvable. Let G = Gertie's age now, and 
J" = Jennie's. By the conditions of the problem, G = SJ. After the 
required number of years, 2J-{- twice the required number of years 
= G -\- the required numxber of years, or 2 J^ + the required number of 
years = G or SJ, Hence the required number of years must be 6 J", or 
6 times Jennie's age now. That is, no matter what Jennie's age may be, 
when it shall have been increased by 6 times itself, it will be J of 
Gertie's. To illustrate ; suppose Jennie's age is 4, Gertie's 32. Jennie's 
age added to 6 times itself ^ 28 years, which is J of Gertie's, the latter 
being 32 + 24, or 56 years. Illustrations might be multiplied ad infinitum, 

"TIME" PROBLEMS, 

75. If J = i, I of the time past noon = | of the time to noon ; and 
f + |, or I, of the time to noon = 24 hours. Hence the time to noon is 
9 hours, and the time is 3 a.m. 

76. f of time past = i oi time to come — 1 hour. 
J of time past = ^ of time to come — i hour. 
J of time past = | of time to come — f hour. 

That is, the time past noon is IJ hours less than f of the time to mid- 
night, f of time to come + f of same — IJ hours = 12 hours = time from 
noon to midnight. Hence | of time to midnight = 12 + IJ = 13i hours, 
and I of time to midnight = f of 13i = 8 hours. Therefore the time is 
8 hours prior to midnight, or 4 p.m. 



168 TABLE BOOK AND TEST PROBLEMS. 

77. At 4 o'clock the hands are at 12 and 4. To overtake the hour 
hand, the other must gain 20 minute spaces ; and after passing the hour 
hand the other must gain 30 spaces ; or 50 in all. To gain 1 space 
requires lj\ minutes, and to gain 50 requires 50 x lyV = ^^ji minutes. 
Hence the time is 54j^j- minutes past 4 o'clock. 

78. In a true minute the hour hand moves over ii of j^ = xW ^^ ^ 
minute space. In the same time the minute hand moves over i^ of a 
space. Hence the minute hand gains if — -f^^ = iff of a space in 1 true 
minute, and in 20 true minutes it v^ill gain 20 x |ff = 17^ spaces, the 
number the hands are apart. 

79. They are 20 minutes apart twice, (a) At 6 o'clock the hands are 
30 minutes apart, and the minute hand must gain 10 minutes: hence 
10 X ^f = 10 if = the number of minutes it must travel ; therefore the 
time is lO^f minutes past 6. (&) At 32^^^ minutes past 6 o'clock the 
hands are together, and the minute hand must gain 20 minutes, to do 
which it must move over 20 x ^f = 21^j spaces : hence the time is 
32^8_ _j_ 21^9- = 54y6_ minutes past 6. 

80. On a clock face there are 60 minute spaces. Let s denote the 
number of spaces between the two hands. If they exactly change places, 
the hour hand will move through s spaces, and the minute hand through 
60 — s spaces. But the minute hand goes over 12 spaces while the hour 
hand goes over 1 ; that is, their rates and distances are as 12 to 1. Hence 
12 : 1 : :60 — s:s, whence s = 4ij\ spaces. At bj\ minutes past 1 the 
hands were together. Since that time the minute hand has gained 4j\ 
spaces, which is ij of the distance it has traveled. Therefore the minute 
hand has moved 4j\ x |J = Jff spaces since 5j\ minutes past 1. Then 
the time was 5j\ + J|f = 10j\% minutes past 1 = 10 minutes 29^^ 
seconds. 

81. First Solution. At 5 o'clock the minute hand is at 12, the other 
at 5. As the former moves 12 times as fast as the latter, the distance 
traveled by the latter in any given time is Jj- of the gain of the former. 
In this instance, to overtake the hour hand, the other must gain 25 
minute spaces : hence the hands will be together at if x 25, or 27j\ 
minutes past 5. The hour hand is then 2^^ spaces from 6. Half the 
gain from then until the required time is the distance each hand is from 
6. The gain is 11 times the distance traveled by the hour hand. Then 
5} times that distance + that distance, or 6} times the same, is 2^j 
spaces; and j\ of |J = j^a* "the distance" moved by the hour hand. 
2j\ — j\^s = 2j\ spaces, the distance of hands from 6. Hence 30 + 2/j 
= 32j\ minutes past five, the time. 



ARITHMETICAL SOLUTIONS. 169 

Second Solution. At half-past 5 the minute hand is at 6, and the other 
midway between 5 and 6. The minute hand moves 12 times as fast as 
the other : hence, by the conditions of the problem, the distance between 
each hand and 6 is 12 times the distance the hour hand moves after half- 
past 5 ; that is, the hour hand has traveled J3 of the distance to 6, or yL 
of 2 J spaces = ^^ spaces ; and the minute hand ^^ of the distance to 12, 
or J3 of 30 spaces = 2^^ spaces. The time, therefore, is 32 minutes ISj^j 
seconds past 5. 

82. Since the minute hand moves 12 times as fast as the hour hand, 
and since in changing places both hands move just once around the dial, 
the distance between the hands at first is -^\- of the dial's circumference 
(60 spaces), or 4j\ spaces. Since 2 o'clock the minute hand has gained 
11 times the distance the hour hand is past 2, or 10 + 4j\ minute spaces. 
Then -^j of 14 j\ = 1 iVs' "^^^ number of spaces the hour hand is past 2 ; 
and 1 j-^/3 + 43-83 -f 10 = loifl, the number of spaces the minute hand is 
past 12, or the time required is 15 minutes 66j^j2_ seconds past 2. 

83. (a) It is evident that the hour hand will be the first to be midway 
between the others, and that it will be there inside of a minute. Suppose 
the hour hand moves s spaces : then the minute hand moves 12 s, and the 
second hand 720 s. For convenience denote the hands by H, M, and S. 
Now, from II to M is 12s — s = lis. Since 720s is less than a minute, 
60 - 720s is the distance from S to 12. Then 60 - 720s + s is the dis- 
tance from S to Hj which, by condition, must be the same as from H 
to M. Hence 60 — 719 s = 11 s, or 730 s = 60, s = j\ = the part of a min- 
ute space the hour hand has passed over. j\ x 12 = |^f = distance 
minute hand has gone. Then f | of 60 seconds = oO-ff seconds, the 
required time. 

(5) When S passes JET, it will directly be midway between H and M. 
This will occur in less than 2 minutes after 12. Suppose H moves over 
s spaces : then iY moves over 12 s, and S over 720 s. Then 

720 s - 60= distance from 12 to S (1) 

Trom IT to ilf is 12 s — s = 11 s spaces. Therefore from ZT to S is ^J- s, 
and s + -y^s = distance from 12 to >S^ (2) 

We have now 720 s — 60 = s + -y- s, whence s = i^4^2V* 

j\%\ X 720 = -\%%Y = ^0i^¥2T seconds, 
the time till S is midway between H and M. 

(c) When the second hand passes J/, it will soon be far enough to 
leave M midway between itself and H. This will occur in a very few 



170 TABLE BOOK AND TEST PROBLEMS, 

seconds. Let s be the number of spaces Amoves from 12. Then iff goes 
12 s spaces ; and S, 720 s, 

720 s ^60^ distance from 12 to S (1) 

and lis =s distance from ^Tto M. After the first minute, S moves over 

s + 11 s + 11 s = 23 s := distance from 12 to S (2) 

Comparing (1) and (2), we see that 720 s — 60 = 23 s, or s = /g^y, which 
reduced to seconds gives 61f |f seconds, the required time. 

GENERAL ANALYSIS. 

84. If 40 pounds contain J a pound of salt, to contain IJ pounds will 
require 40 x 3 =; 120 pounds. Then 120 -- 48 =;= 72 pounds = amount to 
be added. 

85. In the same time B does 4J -^ 3| ^ 1|- times as much work as A. 
A works 4 hours, and B in 4 hours does as much asAin4x 1J = 5 hours. 
Hence what both did in 4 hours could have been done by A in 9 hours. 
Then 9 + 4J == 13| hours, A's time ; and f of 13^ = lOf hours, B's time. 

86. A's + B's + C's = $ 150 (1) 
A's - B's -h C's = .^50, by first condition. 

Hence 2 B's == $ 100, and B's = $ 50. 

Taking B's from (1), we have 

A's + C's = $100 (2) 

By the second condition, 

A's + B's ~ C's = J C's, 
or A's - C's = J C's -B's, 

or A's-|C's=-$50 (3) 

Comparing (2) and (3), we find that | A's = $ 100, or A's = $40. Then 
(2) becomes $40 + C's = $100, whence C's = $60. 

87. In 14 days the men first employed would do J x |f, or j\ of the 
work. Then ^ — j\ = j\^ what 20 men do in 14 days, and in 1 day they 
do i\ of j\ = 2^2 of ^^® work. 1 man does ^V ^^ 2! 2 = tuV^ ^^ ^ ^^y? ^^ 
all the work in 1008 days. He does ^ of it in 336 days. Hence to do ^ 
of it in 12 days requires 336 -~ 12 =: 28 men, the number employed at first. 

88. Call the broken part B, and the stump S. Then 2 (5 + 1 foot) 
= S-i, ot2B-^2 = S-4, ot2B + Q=S. But J5 + iS = 33 feet. 
Using the value of S, we have 7:? + 2 J5 + 6 = 33, or 3 ^ = 27, and B = 9, 
the broken part. 2 L' + 6 = 24, the stump. 



ARITHMETICAL SOLUTIOJSIS. 171 

89. In 1 day A and B can do -^^ of the work. 
In 1 day B and C can do J^ of the work. 
In 1 day A and C can do jL of the work. 

Adding these, we see that A + B + C can do J ( J^ + jL -{- -f^)=^ of 
the work in 1 day, and to do f will require | -^ J = 8 days. 

P A 

90. 10-f SO cost $11.00 (1) 
20 + 10 cost $ 13.00 (2) 

30 + 30 cost $24.00 

or 10 + 10 cost $ 8.00 

But 10 + 20 cost $11.00 

10 cost $ 3.00 

1 cost .30 

Substituting this value in (1) or (2), we find that P cost 50 cents. P 
represents a bushel of potatoes, and A a bushel of apples. 

91. We observe that A's financial ability was twice B's, and that B's 
was twice C's. Hence A's was 4 times C's. 4 times C's — C's = 3 times 
C's = $ 12,000, and 

C's share of cost = $4,000 

A's share of cost = 16,000 
B's share of cost = 8,000 

Total cost = $28,000 

92. Fast boat crosses in 4 minutes, and is ready to start back in 6 
minutes, when the slow boat is just in the middle of the river. They 
now travel towards each other, and the two together must travel half the 
width. Since the fast boat travels 3 times as fast as the other, it will 
travel f of the distance, or | of J = f of the width. To do this will re- 
quire f of 4 minutes =1^ minutes. Hence 6 + IJ = 7J minutes = time 
from start till first meeting. 

93. Every cuck is three-ply, and contains J x 3 = | of an inch of cloth. 
For every tuck there is a single ply ^ of an inch wide. Every tuck, there- 
fore, requires | + J = | of an inch of cloth. 1 yard = 36 inches, and 
36 -r- 1 = 41, the number of tucks. 

94. If f were stolen, | remained, f of (| + $60) = | + $40 = amount 
spent. (3 4. $60) - (f + $40) = J- + $20 = remainder. ^ + $20 + $ 10 
= amount after getting $10. ^ of (i- + $30) = -^^ -]- $6 = amount lost. 
J + $30 - (/o + $5) = J + $25 = remainder, which is half the original 
sum. ^ + $25 = J, or I : hence $25 = |, or i, of his money, and $75 
= what he had at first. 



172 TABLE BOOK AND TEST PROBLEMS. 

95. By the conditions, 2 (broken part + 2) = stump — 8, or (2 x broken 
part) + 4 = stump — 8. 2 x broken part = stump — 12. Then 6Q — 12 
= 54 = 3 times broken part, and broken part = 18 feet. 66 — 18 = 48 
= stump. 

96. The cow cost the least. The cost of an ox = twice that of cow 

— ^12. The cost of horse = 4 times this = 8 times cost of cow — ^48 

— $40. Hence 11 times the cost of cow - $100 = $350. 11 times the 
cost of cow = $450, whence the cow cost $40iJ, ox $69j\, and horse 
$239j-V 

97. I can pick twice as many as I can dig : therefore I must dig twice 
as long as I pick. Hence I must dig 4 days and pick 2. Then 20 x 4 
= 80 bushels, the required number. 

98. For pasture, 1 cow requires J an acre, and for roots ^ of an acre; 
for both she requires J + J = f of an acre. 20 -^ § = 30, the number of 
cows. 30 -T- 2 = 15, the number of acres in grass. 

99. The head and tail are 6 inches + J of the body. The body is the 
same. Hence 6 inches must be | of the body, and 8 inches the body. 
The tail is 3 inches + 2 inches = 5 inches ; and 8 + 5 + 3 = 16 inches, 
length of fish. 

100. The cost of the watch + J its cost = f of its cost, which is $20 
more than both cost, or $120. J = i of $120, or $40 ; and |, or cost of 
the watch, is $80. 

101. A, the more rapid walker, determines the time of the race. 
Therefore the second time was f of the first ; and f x j^^ = |^ of B's first 
distance — his first distance = -^^ of the same, or 54 feet. Then |§, or 
B's distance, was 54 x 80 = 4320 feet ; and 4320 + 80 = 4400 feet, the 
length of the course. 

102. By the conditions of the problem, equal weights of gold and 
silver are in volume as 1 to (-|f^ -^ 20) = }^. Now, if gold w^ere sub- 
stituted for silver in the given volume, which has 2 units of weight and 
21 units of value, there would be 1 + ff^ = ^||- units of weight; 
and 49 6 X 20 = iff^ units of value. Then iffA ^ 21 = 2^1 4^ the re- 
quired number of times. 

103. The italicized clause may be differently interpreted. This solu- 
tion assumes that B received 3s. 9d. less than he would have received had 
he and A finished the work in the specified time, 5 days. For the other 
interpretation, see '^ Algebraic Solutions." For f of the work A should 
receive f of the 90 shillings, or 50 shillings; then B and C would receive 
90 — 50 = 40 shillings, of which B receives 40 — 3J = 36} shillings, for 5 



ARITHMETICAL SOLUTIONS. 



173 



days/or 7 J shillings per day. Hence 90 ~ 7 J = 12if days, the time in 
which B could do the work. 3J -t- 2 = If shillings, C's wages for 1 day. 
Then 90 -i- 1| = 48 days, the time in which C could reap the field. 

104. 117 -^ 1.4 = 81, which is the square of the smaller number. Hence 
Vsl = 9, the smaller number ; and 9 x 1.4 = 13, the larger. 

105. Before meeting, the slower train travels the shorter distance; 
after meeting, it travels J 



the longer distance, while the faster train 
Hence shorter distance : longer distance : : J longer 



travels the shorter, 
distance : shorter distance, or shorter distance squared = J longer distance 
squared, or square of longer distance = 4 times square of shorter dis- 
tance. But the square of any number is equal to 4 times the square of its 
half. Hence shorter = J longer, and rates are as 1 to 2. 



or 



106. 

Hence 

Then 



107. 

and 

Hence 
and 

108. 



and 
109. 



MENSURATION. 
The Bectangle. 

I its perimeter = 100 rods. 

Length = IJ x width. 
2| X width = 100, 
. width = 40. 
length = 60. 
Area == 40 x 60= 2400 square rods. 
2400 -^ 160 = 15 acres. 

1344 -(4 X 3)= 112, 

Vll2 = 10 + = ratio of sides. 
4 X 10 = 40 = length, 
3 X 10 = 30 = width. 



Hence 



I - I = A = 35 gallons. 

2 4 _ 35 X 2^ = 120 = contents in gallons. 
120 X 231 = 27,720 = contents in cubic inches. 
10 X 2| X 144 = 3960, 
27,720 -f- 3960 = 7 feet, the required length. 

80 X 80 = 6400 square feet = area of garden. 
\ of 6400 = 1036 J = area of the walk. 
6400 + 1066f = 7466| = area of square including both. 
V7466| = 86.4+ = side of square including both. 
86.4 - 80 = 6.4 = twice the width of walk. 
6.4 -^ 2 = 3.2 feet = width of walk. 



174 TABLE BOOK AND TEST PROBLEMS. 

110. 45 -4- 30 = 1.5 = area of square end. 

Vl.5 = side of square. 
Then \/l.5^ + vT^ = square of diagonal, 

and \/3 = diagonal = diameter of log. 

111. For convenience put s = the side. Then the diagonal is y/s^-\-s^=: 
s\/2 = 1.4142 X s. By the problem, 1.4142 s - s, or .4142 s = 8.284 rods. 
Hence 8.284 -f- .4142 = 20 rods, the side. The area is 20'^ = 400 square 
rods, or 400 -f- 160 = 2\ acres. 



112. The diagonal of base = Vb^ + 6^ = 10. 

The altitude = VlS'^ - b'^ = 12. 

Volume = 6 X 8 X -y- = 192 cubic units. 

Let 2Xf 3 03, and ix — the dimensions of the rectangular solid : then its 
volume = 24^5, which, by condition, is equal to 192. Hence x^ = 8, and 
X = 2 ; and the sides of the solid are 4, 6, and 8. 

113. Since their rates are as 5 to 6, their distances are as 5 to 6, and 
one runs 26 yards farther than the other: hence 26 x (5 -|- 6)= 286 = 
perimeter of field, and half the sum of the length and width = -if-^. 
Half the difference of the same is •^/[(if^)^ — 4840 yards] ^-^^ yards. 
/. if ^ + -3/ ,^ 88 yards, the length ; and if^ - -^3- = 55 yards, the width. 

114. 8 gallons = 1848 cubic inches. Assuming the dimensions of the 
smaller can to be 3, 7, and 11 inches, the contents would be 231 cubic 
inches. By similar solids, we have 231 : 1848 : : 3^ :x^ {= 216), whence x, 
or end of larger box, is 6. Then the other sides are 14 and 22. Since the 
second can requires 4 times as much tin, it has 4 times as much surface ; 
and, by similar surfaces, its dimensions are Vi = 2 times the first. Its 
contents are 2^ = 8 times the smaller = 8 x 8 = 64 gallons. 



The Triangle. 

115. The saw is the hypothenuse of a right-angled triangle, whose per- 
pendicular is 8 feet. .-. 102 - 8^ = 36 ; and \^m = 6, the width. 

116. The base x J the altitude, or J the base x altitude = area. But 
the area = 972 perches, and ^ the base = 36 perches. Hence 36 x the 
altitude = 972, whence the altitude is found to be 27. Now, either side 
is the hypothenuse of a right-angled triangle whose perpendicular is the 
altitude, or 27, and whose base is 36. Hence VS6'^ + 27*-^ = 45, the length 
of the sides. 



ARITHMETICAL SOLUTIONS, 



175 



The half sum is 



117. The sum of the sides is 9 + 14 + 5 = 28 rods. 
14. Subtracting each side separately, we have 

14- 9=: 5. 
14 - 14 = 0. 
14-5 = 9. 
Multiplying these three remainders and the half sum together, we have 

6x0x9x14 = 0. 

Extracting the square root of 0, we have Vo = 0, the required area ; 
that is, there is no area, and the problem is a " catch." The three given 
sides or lines do not form a triangle, since one is as long as the other two. 
They form one straight line only, or, if not, then two of the sides do not 
meet. 

118. Because it cuts the sides proportionally, it is parallel to the 40-foot 
side, and is therefore half as long as the base, or 20 rods long. 

119. In this problem the height of the pitch is f of 36 = 24 feet. This 
is the perpendicular distance from the vertex of roof to the square, and is 
one leg of a right-angled triangle whose hypothenuse is the length of a 
rafter. Half the width of the building is the other leg. Hence 

\/24'^ + 182 ^ 30 - length of rafters. 

Note. — If the rafters project, add the projection to the length of rafters. 

120. The rope will form the hypothenuse of a right-angled triangle 
whose perpendicular is 10, and whose base is the circumference of the 
cylinder. The circumference is 6 x tt = 18.8496. Then 

the length of the rope = ^[(lSM96y -\- (10)2] = 21.337 feet. 

121. Let ED represent the corner of the garden wall, C the position of 
the object, B the center, and 

BF the height required. ' 
From one corner to the cen- 
ter of the lot is half the diag- 
onal = JV3002 -I- 3002 

= 212.13 - feet = BD. 
CDE and CBA are similar 
right triangles : hence CD : 
CB::DE:BA, or 20: (20 -f 
212.13) : ; 9 : BA ; whence BA 
feet. 




104.45. Then BF 



176 TABLE BOOK AND TEST PROBLEMS. 

122. J sum of sides = 150 : hence area = \/(150 x 503) _ 4330 -}- square 
rods. J the area == 2165 rods, which is a sextant, or J of a circle, because 
each angle of the triangle is 60°, or J of a circle. The entire circle = 2165 
X 6 = 12,990 = 7rr2. 12,990 -- 3.1416 = 4134.83 = r'^. V4 134.83 = 64.3 = r, 
the length required. 

123. Assume a similar room whose side is unity, or 1. Its diagonal is 
V2 ; and from the middle or center of floor to one corner is J V2. There- 
fore the distance from the center to an upper corner is 

V(lV2)2+l2:= VI..1V6. 

By similar triangles, then, we have jV6:24: : 1 :»:, the required side, 
whence x = 19.596 feet. 

124. By the first condition, one tree is 80 feet high ; and by the second 
condition, the other is 50 feet high. One tree is therefore 30 feet higher 
than the other. This 30 feet is the perpendicular, and the width of the 
river is the base of a right-angled triangle whose hypothenuse is the dis- 
tance betw^een the trees' tops. Then 30'-^ + 40^ = 2500 = square of the 
hypothenuse, which is 50 feet. 

125. Since the shortest distance between two points is a straight line, 
the fly must travel along the hypothenuse of a right-angled triangle. One 
leg of this triangle is the length of the room, and the other is the width 
+ the height. Hence (16)"-2 + (12 + 8)'^ = 656 = the square of the hypoth- 
enuse. The fly must therefore travel \/656 = 25.61 + feet. 

Note. — This is clearly seen by setting up four small boards to represent the sides 
and euds of the room, then laying down the boards without displacing them, and drawing 
a line from the starting point of the fly to its destination. 

126. The diameter of the log will be the diagonal of the log when 
squared. Hence it will be the hypothenuse of an isosceles right-angled 
triangle. One leg of this triangle, or one side of the squared log, is found 
by taking the square root of J of the square of the diameter of the round 

log. In this case it is v ^(2 V2)"^ = \/4 = 2 feet = 24 inches. Therefore 
one board contains 16 x 2 = 32 feet. To find the number of boards, 
divide 24 by 1 -f \, which gives 19+ (if nothing were wasted by the width 
of saw, there would be 24 boards). Therefore 32 x 19 = 608 board feet. 

Notes. — 1. This method of finding the number of boards is not exactly correct, since 
there is one more board than saw-cut, provided there is no loss of a thin board at the last. 
Then, too, it is possible to cut some boards from the slabs cut off in squaring. We have 
considered only the squared log. 

2. Arithmetics give the following rule for finding the side of the inscribed square, but 
do not explain: "Multiply the given diameter by .707106." In the given problem, 
2v^2 X .707106 = 2, same result as obtained in a different way. 



ARITHMETICAL SOLUTIONS. 



177 



127. Let BCDE be the board. We have given BG = 4 inches, or 
i of a foot; ^7= 12 feet; and EF=S inches, or f of a foot; also 
EI= 2 (16 — 8) = 4 inches, or | of a foot. Since the trapezoid decreases 



R 



n_ 



B 



E 



8 inches in 12 feet, it will decrease 16 inches (that is, come to a point) 
in 24 feet. Hence AF= 24 feet. Then 24t - 12 = AG = 12 feet. The 
area of the board BCDE is 12 square feet. Area of EFHK (J of the 
board) = 3 square feet. The area of the triangle EFA = | x 12 = 8 
square feet. The area of EFA - EFHK rr 8 - 3 ^r 5 ^ area AHK, 
Then, by similar surfaces, we have EFA : KHA ; : AF^ : AH^, or 
8 : 5 ; : 676 : (x = 360). Hence AH = VSQO = 18.97+ feet. AH - AG 
= GH, or 18.97 - 12 = 6.97, the distance from narrow end. 12 - 6.97 
= 5.03 feet = distance from wider end. 



Tlie Circle. 



128. Let AB be the diameter of the floor, and AD = BE = width of 
granary. Put 5(7 = 15 = r, and CE =19 = R, Then irr^ = area of inner 
circle, or floor ; and irB'^ = area of outer 

circle, or floor and granary ; and 

wR'^ — 7rr2 = area of granary 

= 7r(i?2 _ r2)= 7r(361 - 225) 
= 136 TT = 427.2576 square feet, 

which multiplied by 3 = 1281.772+ cubic 
feet = 1030 bushels, the contents of the 
granary (in standard bushels). 

129. 160 acres = 160 x 160=25,600 square 
rods = area. 25,600 - ^ 3.141 6 = 8148.7 = 

square of radius. V8148.7 = 90.2+ = radius of field. If } is plowed, 
80 acres remain in circular form. 80 acres = 12,800 square rods. 
12,800 -4- 3.1416 = 4074.35 = r2 of remainder. V4074.35 = 63.8 rods = 
radius of remainder. 90.2 - 63.8 = 26.4 rods = width of ring plowed. 
26.4 rods = 5227.2 inches, which divided by 18, the plow's width, gives 
290.4 = number of rounds. 

ellwood's test prob. — 12. 




178 



TABLE BOOK AND TEST PBOBLEMS. 



130. The wheels describe two circles, one within the other. Circum- 
ferences vary as their radii : hence, since the outer circumference is twice 
the inner, its radius must be twice the radius of the inner. But the dif- 
ference between their radii is 6 feet, which must be the less radius, and i 
of the greater. Then the diameter of the inner circle is 6 x 2 = 12 feet, 
and its circumference is 12 x 3.1416 = 37.6992 feet. 



131. Let h — the man's height, and r — radius of the earth. Then 
2 Trr = circumference of the earth, and r + /i = radius of circle described 
by the man's head. Then 2 7r(r + li)— distance traveled by the top of 
his head ; and (2 7rr -t- 2 irli) — "I-kt — distance his head travels farther 
than his feet = 2 ttTi. That is, in any such problem, to find how much 
farther the head (or top) travels than the feet, multiply the height of the 
man (or object) by 2 tt, or 6.2832. In this problem we have 6 x 6.2832 = 
37.6992. 

132. Let C(^-CF-CE-r- the length of the rope, and C the point 

where it is fastened. Since the angle BCD 
is a right angle, it cuts off \ of the circle : 
hence, CFGE is | of the circle. If 10 acres 
is f, the circle contains 10 x | = 13J acres 
= 2 133 J- square rods = irr^. Dividing by tt, 
and extracting square root, we have r=:26 + 
rods, the length required. 

133. First Solution. The thickness makes 
no difference so long as it is uniform. The 
area of one side is 7.0686 square feet = 
1017.8784 square inches, of which A grinds 
off I, B 2V1 and C \. After A and B have ground, C has his \ left in 
circular form. Then 254.4696 -t- tt = 81 = the square of the radius of C's 
circle. Hence, V8l = 9 = C's number of inches. B's share = v^ = 
356.2574 square inches ; and B's share + C's share = 610.72704 square 
inches, which is also in circular form. The radius of this — the square 
root of (610.72704 -^ 3.1416)= 13.94 inches. This is C's radius + B's 
radius: hence B's share = 13.94 - 9 = 4.94 inches, and A's must be 
18 - 13.94 = 4.06 inches. 

Second Solution. After A grinds, f of the stone remains for B and C. 
Since the radius is 18 inches, and similar surfaces are to each other as 
the squares of like dimensions, the radius of this remainder is Vf x 18 = 
13.9428 inches. Hence A grinds off 18 - 13.9428 = 4.0572 inches. Since 
C owns \ of tlie stone, the radius of his share is V^ x 18 = inches. 
Then B must grind off 18 - (9 + 4.0572) = 4.9428 inches. 




ARITHMETICAL SOLUTIONS. 179 

134. Let AB = 20 = 2m, the length of the given line, H = radius of 
larger circle, and r = radius of smaller circle. 
In the right-angled triangle ADC we have 
AC = R, DC = r, and AD = m ; whence 
jfj2 _ |.2 _ ^i2 Multiplying this equation by tt, 
we have wR^ — irr'^ = wm^. Since ttR^ = the 
area of the larger, and irr^ = the area of the 
smaller circle, the left member expresses the 
difference between the areas of the two circles, 
or the area of the ring or track. Then w^n'^ = 
3.1416 X 10^ = 314.16 square rods, the area of 
the ring. The second requirement is indeter- 
minate. If r = 0, i? = 10, which is the least value R can have. Eor any 
value of R less than 10, the value of r is imaginary. Suppose R = 6: 

then VW^^^wP- = VS6 — 100 =:V— 64=:8V— 1. R may have any value 
from 10 to GO : then the width of the track would vary from to 10 rods. 

Pyramids and Cones, 




135. Slant height = V12M-5^ = 13. J slant height = 6.5 feet. Cir- 
cumference of base = 10 x 3.1416 = 31.416 feet. Then 31.416 x 6.5 = 
204.204 square feet = 22.689 square yards. 

136. The top section was a cone whose volume was \ of the loaf. The 
top section and the next one, taken together, formed a cone whose vol- 
ume was f of the loaf. Hence, observing that similar solids are to each 
other as the cubes of their like dimensions, we have the following pro- 
portions : 3 : 1 : : 203 : 7^3, whence /i = 20 \/I = -'/- v^9 = 13. 86722 + inches, 
the height of the top section; and ^:2:\20^:K^^ whence h = ^^-\^ = 
17.4716+ inches, height of both upper sections. Then 17.4716 - 13.86722 
= 3.60438 inches, height of middle section ; and 20 - 17.4716 = 2.52839 + 
inches, height of bottom section. 

137. The bucket is a frustum of a cone, and its volume = tt (i?2 _^ f^ _|_ ^,.) ^^ 

o 

in which a = altitude, R = radius of larger base, and r = radius of smaller 
base. The areas of the bases vary as the squares of the radii. Hence, by 
condition, jR^ir^: ;5:3, whence SR^ = 5^2, or R^ = f r^, and R = rVf ; 
also i?r = r^Vf. Substituting values in the formula given above, we have 
volume = 3.1416 (r^Vf + r^ + f r2) x i of 12, or 3.1416 (f r^ + r^ + r'^V}) 
x 4 = 13 gallons = 3003 cubic inches. Multiplying, etc., 49.637 r^ = 3003, 
whence r = 7.77 inches, 2r = diameter = 15.54+ inches, R = rVj = 
7.77\/| = 10.03 + , and 2R = D = 20.06+. 



180 



TABLE BOOK AND TEST PROBLEMS. 



138. Altitude of cone completed = 100 feet. Area of larger base = irr'^ 
= 1256.64 square feet. 1256.64 x 33^ (i of altitude) = 41,888, contents 
of cone. 41,888 — 12,000 = 29,888, contents of cone after frustum is cut 
off. Since similar cones are to each other as the cubes of their altitudes, 
we have 41,888 : 29,888 : : 100^ : h^ ; whence h = 89.9+ feet. Then 100 - 
89.9 = 10.1 feet, the length taken off. 

139. Let r = radius of upper base. 
Then 2 r = radius of lower base, 

7rr- = area of upper base, 

4 7rr^ = area of lower base, 

and \/4:7rf^ x irr^ = 2 Trr^ = area of mean base. 

7 wr^ = sum of the three bases, which multiplied by J of the altitude 
= the volume of frustum. Hence Ittt'^ x (J of 12)= 287rr^ = 7050 cubic 
inches, and Trr^ = 251.7857 + . Dividing by tt = 3.1416, r^ = 80.1457 + , 
whence r = 8.95 + , and 2r = 17.9 inches, the diameter of upper base. 
Then 17.9 x 2 = 35.8, the diameter of lower base. 

140. The bucket, represented by ABCD^ is a frustum of a cone whose 
volume = contents of bucket = 265.98 cubic inches. J of 265.98 = 88.66 

= volume of water = EFGD. Completing the 
cone, its altitude is 56 inches, and its solidity 
B = ttR^ X -V- = 718.38. Then 718.38 - 265.98 
= 452.4 = volume of CDK, and 452.4 + 88.66 
GT^^ P+ Z^R = 541.06 = volume of EFK. Since similar 

cones are to each other as the cubes of their 
altitudes, we have 718.38 : 541.06 : : 56^ : S^ 
whence KI = 50.9+ inches. Then 50.9 + 3 
(the rise) = 53.9 = KP, and we have the pro- 
portion 53.9 + ^ : 56^- ; GHK : 718.38, whence 
GHK = 642.3 + cubic inches. Then the volume 
of GHK — the volume of EFK = the volume 
displaced by the ball. Hence 642.3 -541.06 = 
101 .24 = volume of the portion of ball submerged. 

141. The diagonal of the largest square inscribed in the base of the 
cone = diameter of the cone, or 10 feet. Hence J of 10^ = 50, the square 
of one side, or the area of base of pyramid. Then 50 x ^ of 30 = 500 
cubic feet, the volume required. 



^~""-N 


-. 


_^ 




/ 


V^^---^ 


~ZZ^^ 


\ 


/ 


eVch 


HH^ 


. \ 




cCI3 


— ^-^ 


\\ 


K 



Similar Solids. 

142. The dimensions of similar solids are proportional to the cube roots 
of their volumes. Hence v8 : Vl : : 18 J :x::S:y ; whence x = 37, the 
width ; and ?/ = 16, the depth. 



ABITHMETICAL SOLUTIONS. 181 

143. The water forms a cone similar to the glass. Hence, by similar 
solids, J : 1 : : /i^ : 33. whence h = V6.75 = 1.8+ inches, depth of water. 

144. By similar soUds, 180 : 1015 ;: (5f)3 : /iS . whence ^ = lOff = 10 
feet 4.6 inches. 

145. Since spheres vary as the cubes of their diameters, the diameter 
of a ball containing as much as the three is the cube root of (3^ + 4^ + 5^) 
= v^^iezrC. 

146. Consider the 10-inch shell solid. Then its weight is found by 
similar solids, thus: 8^:103::8 pounds : (x = 15| pounds). Since the 
shell weighs 7| pounds, we have 15f — 7f = 8 pounds = the weight of a 
ball just filling the hollow part : but this is the weight of the given ball, 
whose diameter is 8 inches. Hence (10 — 8) -r- 2 = 1 inch = thickness 
of shell. 

Cubes and Spheres. 

147. The diameter of the water = 12 — 2 = 10 inches. The contents 
of the water = 10^ x .5236 = 523.6 cubic inches. Then 523.6 h- 231 = 
2.26| gallons. 

148. 36 X 16 X 3 = 1728 cubic inches = volume of water raised, which 
is also the solidity of the cube. Hence ViT2S = 12 inches = the edge of 
the cube. 

149. The solidity of a 6-inch cylinder 4 inches long = 3^ x 3.1416 x 4 = 
113.0976 cubic inches, which must be the volume of the sphere. Volume 
of a sphere = d^ x Jtt. Hence 113.0976 = .5236 #, whence # = 2I6, and 
d = 6 inches = diameter required. 

150. The volume of any sphere equals two thirds of its circumscribing 
cylinder. In this problem the submerged ball occupies two thirds of the 
bucket : hence the bucket is just as deep as wide ; that is, the bucket is a 
cyUnder whose length and diameter of base are each equal to the diameter 
of the ball, or 10 inches. 

151. The diagonal of a cube = its side x Vs. The side of this cube, 
then, is 10 -^ Vo = 5.773, and its contents = 5.773^ = 192.4 cubic inches. 
Since the cube's diagonal is the sphere's diameter, we have the volume 
of sphere = 10^ x ^ir = 523.6 cubic inches. Hence the contents required 
= 523.6 - 192.4 = 331.2 cubic inches. 

152. Sohdity of ball = 3^ x ^tt = 14.1372 cubic inches, and J of this = 
3.5343 = the share of each. After the first three have wound their shares 
off, the last one has a ball containing 3.5343 cubic inches, whose diameter 
is the cube root of (3.5343 -r- .5236) = 1.89, the last one's share of the 



182 TABLE BOOK AND TEST PROBLEMS, 

diameter. After the first and second liave done, the last two have a ball 
containing 7.0686 cubic inches, and its diameter is the cube root of 
(7.0686 -- .5236) r= 2.38 + . Then 2.38^ 1.89 = .49 = third lady's share. 
After the first lady has wound, the last three will have a ball whose con- 
tents is 10.6029 cubic inches, and whose diameter is the cube root of 
(10.6029 -- .5236) = 2.72. Then 2.72 - 2.38 = .34 = second lady's share, 
and 3 - 2.72 = .28 = first. Hence the first winds off .28, the second .34, 
the third .49, and the fourth 1.89. 

153. Will the sphere be totally or only partially immersed ? In order 
to determine this, we construct the diagram, in 
which ABC represents a vertical section of the 
glass, and FGH a section of the sphere. By the 
conditions, DB = 3, EF= 2, and i>C = 8. BO 
= V8"-^ +3^ =\/73. Now, by similar triangles, 
DB:BC::EF:EC, or 3 :\/73 : : 2 :x = |\/73 = 
6.696+ inches. DE= nC-EC=S- 5.696 = 
2.304 inches. But, E being the center of the 
sphere, EH is a radius = 2 inches. Therefore 
ED is greater than EH, and the sphere is totally 
immersed. Hence the volume of water displaced 
is equal to the volume of the sphere, which is 
represented by | irr^ = 33.51 cubic inches. 



MISCELLANEOUS PROBLEMS. 

154. We invert to find how often the divisor is contained in 1. We 
then multiply by the number of times 1 is contained in the dividend, 
which gives the required quotient. 

155. In such expressions the operations indicated by the signs -r- and 
X must be performed first. Hence the given expression is the same as if 
written thus, 15 + (9 -^ 3) - (2 x 3), and its value is 15 + 3 - 6 = 12. 

156. The log's solidity = 20 x 3 x 2^ = 150 cubic feet. A cubic foot of 
water weighs 62J pounds. Hence a cubic foot of oak weighs 62J x .925 
= 57.8125 pounds, and 150 cubic feet weigh 150 x 57.8125 = 8671.876 
pounds. 

157. U.'s chance of succeeding is j?, and his chance of failing J. B.'s 
chance of success is y\, and his chance of failure yV. Hence probability 
(1) is I X fi= /j; (2) is i X A = /s; (3) is f x /j = A ; (4) is 




ARITHMETICAL SOLUTIONS. 183 

158. Let r = rate of increase, p = population, t = the time. We find 
r = 6^0 — yV = ^1 • ^^ *^^ ^^^ ^^ ^^^^ y ^^^ ^^^ population is 20 -hrp=p (I -\-r), 
Sit the end of second year it is p(l + r)'^; and so on till the end of the 
t years, when it is 2^0- + 0*- Substituting values, we find the population 
to be 6,000,000 (1 + ^Jo)^^'^ = 10,206,357. The increase = 10,206,357 
- 5,000,000 = 5,206,357. 

159. The first buyer they met was paying 1 cent for 7 eggs. At that 
price A sold 7 cents' worth, retaining 1 egg; B sold 4 cents' worth, 
keeping 2 eggs; C sold 7 eggs for 1 cent, and kept 3 eggs. The next 
purchaser was a sort of *' bull," and offered 3 cents apiece, at which price 
they all sold out, A realizing 3 cents, B 6 cents, and C 9 cents. Thus 
each got 10 cents for his eggs. 

160. Let A denote the faster train, B the other. Then to pass B when 
going the same way, A must gain 230 + 210 = 440 feet, and 440 -f- 15 = 29|- 
feet, which is A's excess of rate per second. In opposite directions they 
together move 440 feet in 3| seconds, or 117 J- feet per second, of which 
A runs 29i- feet more than B. Then J of (117^ - 29|)=: 44 = B's rate 
per second, and 44 + 29i =: 731, A's rate. (44 x 3600) h- 5280 = 30 (miles 
per hour), and (73} x 3600)— 5280 = 50 (miles per hour), the required 
rates. 

161. 11 4- 9 = 20, which satisfies the first condition. The least multiple 
of 11 that wdll contain 9 with a remainder of 4 is 22. Hence 20 + 22 = 42, 
which satisfies the first and second conditions. The next addition must 
be a multiple of 99, and contain 7 with a remainder of 5. This number 
is 495. 42 + 495 = 537, which satisfies the first three conditions. Similarly, 
537 + 2772 = 3309, the required number. 

162. After J^ has been drawn, j\ remain; -^^ from this leaves 
92 93 94 

AV = ]f^ ^ TO from this leaves ^Wo = ^^'^ to f ^g^ this leaves j%\%\ = — . 

It is thus seen that after any given drawing the part of wine remain- 
ing is expressed by the first remainder raised to the power denoted 
by the number of drawings. Hence, in this case, the part remaining 

QIO 

is -^ = AVoVoWoVo = .3486784401. Then the number of gallons is 
100 X .3486784401 = 34.8678444-. 

163. This is a problem in arithmetical progression, in which a = the 
first term, d = 4, and n = 42. Then I = a -i-(n — l)d = a -{- 164.; and 
s = l(a -\- l)n = 21(2 a -\- 164) = 42 a + 3444 = the whole number delivered. 
By condition, half of these, or 21 a 4- 1722, were delivered in the last 18 
days, (A). Taking the first 24 days, Z = a + 92, and s = 24a 4- 1104, 



184 TABLE BOOK AND TEST PROBLEMS. 

tlie number delivered in the first 24 days, (B). Equating (A) and (B), 
as the condition of the problem allows us to do, we have 24 a + 1104 
= 21 a + 1722, whence Sa = 618, or a = 206. Then 42 a = 8652, and 
8652 H- 3444 = 12096, the number delivered altogether. 

164. Eepresent the names by their initials. Then, by the conditions of 
the problem, S = P + 23, (1) ; and B = A + 11, (2). Since the pieces are 
all squares, we have T>^ + 63, P^ -f 63, and A^ -f 63, each a perfect square. 
Prom a table of squares we find that the only squares whose difference 
is 63 are 1 and 64, 81 and 144, 961 and 1024. Consequently the boys, 
A, P, and D, must be 1, 9, and 31, and the men, S, B, and H, must be 
8, 12, and 32. To satisfy (1), S must be 32, and P 9. To satisfy (2), 
B = 12, and A = 1. Observing the last condition of the problem, it is 
readily ascertained that H and A are father and son : so S and D, and 
B and P. Hence the names of the boys are Adam Hugus, Dick Smith, 
and Paul Brown. 

165. For convenience we call the grass standing on 1 acre a crop, and 

the weekly growth on 1 acre a growth. Then, by the conditions of the 

problem, 

48 oxen in 1 week eat 3 crops and 12 growths (1) 

90 oxen in 1 week eat 5 crops and 30 growths (2) 

Comparing 5 times (1) and 3 times (2), we see that 

30 oxen in 1 week eat 30 growths (3) 

Comparing (2) and (3), we see that 

60 oxen eat in 1 week 5 crops (4) 

Prom (3) and (4) we find that 1 ox in 1 week will eat 1 growth, or ^^ of 
a crop. Now, since the quantity to be consumed is 6 crops and 54 growths, 
it will require 1 ox (6 -^ ^2) + (^^ -^ ^)= 1^6 weeks to consume it. Hence 
the number of oxen required to consume it in 9 weeks is 126 -^ 9, or 14. 

166. 1 square acre = 43,560 square feet, or 208.7 feet square. If 
planted 3J feet apart, 60 hills can be planted in a row, and a space of 2.2 
feet remains. If we divide this space equally among the 59 spaces between 
the 60 hills, they will then be 208.7 -- 59 = 3.537 + feet apart. Now, plant- 
ing in the quincunx order will bring the rows clo ser than 3^- feet, but every 

other row will contain only 59 hills. Then ^(3^)2 _ f^^h±y= 3.02 

feet, the distance the rows are apart. And 208.7 -f- 3.02 = 69 spaces, which 
allow 70 rows to be planted, of which 35 rows contain 60 hills each, and 
35 contain 59 hills each. Therefore (35 x 60) + (35 x 59)= 4165, the 
required number. (From Dr. I. J. Wireback.) 



ARITHMETICAL SOLUTIONS. 185 

167. Taking the quincunx order, we have 12 trees in the first row, 11 
in the second, fourth, and so on. The distance the rows are apart is 
VP — .5'^ = .866, which is the altitude of each triangle. Hence in each 
strip between two rows there is a gain of 1 — .866 = .134 of a rod. To 
gain a row will require .866 -=- .134 = 7 strips. Then 7 + 1 = 8 strips 
contain 9 rows of trees. The odd rows contain 12 x 5 = 60 trees. The 
others contain 11 x 4 = 44 trees. We therefore have 60 + 44 = 104 trees 
in quincunx order. The strips planted cover .866 x 8 = 6.928 rods : 
hence there remain 11 — 6.928 = 4.072 rods, which is not sufficient ground 
to allow the gain of a row by the quincunx order. Hence by that order 
there would be a small loss, and we use the square order, planting 4 rows, 
12 in a row, or 48 trees. Therefore the greatest number is 104 + 48 = 
152 trees. 

168. 90 + 3 + 1 + 5 + f + /s, 9 + 23x^0 4-671, or 15 + 36 + 47 = 98, 
and 98 + 02 = 100. 



186 TABLE BOOK AND TEST PROBLEMS. 



ALGEBRAIC SOLUTIONS. 
FACTORING. 

169. This may be written - (25 x ± 40 Vi + 16) , or - (5 Vx ± 4)2. 

170. The factors of — 20 are 4 and — 5, and their sum is — 1. Vo^ 
multiplied by — 1 = — x: hence x^ — x — 20 = {x -{- 4) (x — 5). 

171. x3 - 2 x2 - 3 X + 6 = x3 - 3 x - 2 x2 + 6 = x3 - 3 X - (2 x2 - 6) = 

X (x2 - 3) - 2 (x2 _ 3) = (x - 2) (x2 - 3) . 

1 i i i 1 _5_ 

172. X = x^ X x^ X x^ , or x^ x x^ x x^^. The latter may be written 

\/x X Vx X Vx^. 

173. x3- a22/n_ ^2x2+ x?/«= x2 (x- a'^) -\-yr^(x-a^) = (a:2 4- 2/n) (x-a2). 

174. The form of this polynomial suggests a binomial factor, (x — 3)2 
= x2 — 6 X + 9, which subtracted from x2 + 10 x — 39 leaves 16 x — 48 = 
16 (x - 3). Hence x2 + 10 x - 39 = (x - 3)2 + 16 (x - 3), which may 
be written (x — 3 + 16) (x — 3), or (x + 13) (x — 3), the required factors. 

175. The sum of the factors of — 14 is — 5. Hence the factors are + 2 
and — 7. Therefore the required factors are (x + 2) and (x — 7). 

176. 10 ai- ^t\-20a = \0a{^ + ^^ - 2 ^ 10a(- - ^V- 

V2/'2 x^l V2/ X* y \y x2y 

177. x2-2/2=:(x-2/)(x + 2/). 

jc24-2/2 = (x- V2x^+2/)(x+ V2x?/ + 2/). 



FRACTIONS. 
178. 

x-\ x4- 



X a-\- x I ^"^ ax-\- x--\- aP' 

a a a + X a + x a^ — x^ x^ — a^ 



a a Or ax — x^ — a^ —a^—x^ x^ -p a' 

X X X 

X a — X a — X a — X 

a a 

179. The reciprocal of a quantity is 1 divided by that quantity. Any 
factor may be transferred from dividend to divisor (or from numerator to 



ALGEBRAIC SOLUTIONS. 



187 



denominator), and vice versa, by changing the sign of its exponent. 
Hence the required reciprocal is 

1 _ (a + b) -« _ (a - by 



(a-b)-^ (a - b) -« (a + b)'- 
{a + by-^ 



180. The numerator may be written 

— H — or ^ ^ ' Hence the complex fraction 
1 



, and the denominator 



x^y^ 



(X - yy ^ 

x^ + y^ (x - 2/)* (X* + ?/*)' 
x^2/* 



SIMPLE EQUATIONS. 



181. Adding, 2(x + 2/ + ^) = « + & + c, 

or x + 2/ + ^ = J(a + & + c) 

x + 2/ = a 
Subtracting, ^ = J (5 + c — a). 

(4) - (2) gives y = l{a-b-\-c), 
(4) — (3) gives X = J (a + & — c). 



(4) 
(1) 



182. Let - = one part : then = — = the other part. By con- 

X 5 X 6x ^ ^ 



dition, (8x — 5)-l-l = 5x + x, or 8x — 4 = 6x, whence x = 2. 



Then 



8x 



5x 



n 
lo' 



X 



183. Let X = the number, of days he was idle. T^en a — x = the 
number of days he worked, ex = amount forfeited, and 6(a — x)=: 
amount earned. Hence b{a — x)— cx = d^ or ab — bx — ex = d; whence 
ab — d 



6 + c 



', the required number of days. 



184. Let X = velocity. Then 14 — x = rate of rowing. (14 — x)— f x 
= rate on the return. Hence (14 — f x)10J = 42, whence x = 6. 

185. Let X = the number of first kind, and y = the number of second 
kind. 

Then x + 2/ = c (1) 



188 TABLE BOOK AND TEST PROBLEMS. 

By the first condition, - = part of a crown made by the x pieces, and 

y ^ 

T = part of a crown made by the y pieces. But both make a whole crown. 

Hence ^ + ^ = 1 (2) 

a b 

Clearing (2), bx -\- ay = ab (3) 

Subtracting a x (1) from (3), we have bx — ax = ab — be, whence x = 

^(^-"). Then2/z.c-x = M^-=^. 
b — a b — a 

186. Let X = the number. Then x — 100,000 = the remainder after 
removing the 1, and 10 (x — 100,000)+ 1 = the new number. Then, by 
condition, 10(ic - 100,000)+ 1 = 3x, or 7x=999,999; whence x=142,857, 
the number required. 

187. Let X = time denoted by minute hand. Then x — 35 = number 

X 

of minute spaces it has passed the figure 7. — = the number of minute 

spaces hour hand has moved since 6 o'clock. Then 5 — — = the number 

12 
of spaces it lacks of figure 7. Hence, by condition, 5 — — = a: — 35, 

1^ 
whence x = 36 ^f minutes past 6 o'clock. 

188. Let X = number of ounces of alloy. 

X 

Then — t p = number of ounces of silver. (1) 

X 

and q = number of ounces of copper. (2) 

_^, X , X - pmn — qmn 

Then — \- p -\ q = ^i whence x~ 



m n mn — m — n 

Substituting in (1), — \- p = - —, the silver. 

^ ^ m mn — m — n 

Substitutmg m (2), q= ^ ~ ~ — , the copper. 

^ ^ n mn — m — n 



189. Let X = A's rate per second. 

?/ = B's rate per second. 
Then 3600 x = A's rate per hour. 

= time in which A runs a mile. 



X 



1760-20 _. 1760 . . ^ ,.^. 
30 = , by first conditions. 

y X ' -" 



ALGEBRAIC SOLUTIONS. 189 

The second heat gives the equation ^ — 32 = Equat- 

y ^ 

1760 - 20 ^^ 1760 - 9j\ oo i. r . t» 

mg, we have 30 = ^ - 32, whence y = 5.3 By 

2/ 2/ 

substituting this value of ?/, we find x = 5i^|, A's rate per second. 
5J-f X 3600 = 21,120 yards = 12 miles, A's rate per hour. 

190. Let X = the duty after reduction, and 2y = any number of pounds 
consumed at 6 cents per pound. Then 12 y = the original revenue. 
After the reduction, the number of pounds consumed in the same time is 
Sy, and the revenue is Sxy, By the problem, Sxy -{- -J- (12?/) = 12?/, or 
3x2/ + 4?/ = 12?/, whence x = 2| cents per pound. 

5 ^^ 

191. At 3 o'clock the hour hand was at^l^T, the other at^ Let x = the 
number of minute spaces traveled by the hour hand. Then 12 x = num- 
ber traveled by minute hand. If they are opposite each other, the 
minute hand is just as far past 9 as the hour hand is past 3, or x spaces. 
Hence the minute hand has passed over (45 + x) minute spaces, and we 
have 45 + x=:12x, whence llx = 45, x = 4 Jj, and 12 x = 4Q^^ = the 
time past 3. 

Note. — Problems of this kind are by no means rare in the old •' mental " arithme- 
tics, and may be readily solved by analysis. The reasoning employed in such analyses, 
however, is the same as that in algebra, to which such problems properly belong. 

192. Eepresent the four numbers by to, x, y, and 0, and let their sum 
= m. Then, by the conditions, we have 

^^m^2Jl=a,orio = 2a-m (1) 




3(2 — m 



U/, 










2 




a, 


or 


y 


= 


4a 


3 


m 


a, 


or 


z 


= 


ba 


4 


m 






m 


^ 


73, 


a - 


-25m 



(2) 

(3) 



s + !Ii_^^ = a,OT e = ""' ^ ■" (4) 

o 

By (1) + (2) + (3) + (4), ^^ 

or 37 m = 73(2, and m = If a. 

Substituting this value of m in (1), (2), (3), and (4), we have 

iv = 2a — ^^^a =-^\a. 

y =i(4«-|?a)=f?a. 
z = J(5a- Jf-a)=|fa. 



190 TABLE BOOK AND TEST PBOBLEMS. 



193. Let X = the distance. 

Then — = the time going, 

and 2 = the time returning. 



{"'^y 



the distance = x. 



Clearing, 96 — 4x = 12x, 

whence x = 6, the required distance. 

194. Let X miles per hour = strength of wind, 
and y miles per hour = strength of tide. 

Since the rates are as 5 to 3, the times are as 3 to 5. Hence j of 
12 = 4 J hours = time out, and f of 12 = 7J = time in. 

By first statement, 60 -f- 4J - 13J = x + ?/ (1) 

By second statement, 60-^7h = S=:x — 2/ (2) 

Adding (1) and (2), we find x = 10|. 
Then tj = 10'l-S = 2|. 

195. For convenience let x = John's age 3 years ago. Then x + 6 
= John's age 3 years hence, and 15 x = the father's age 3 years hence. 
By the first condition, 5 x + 25 = Smith's age 2 years ago. Then 5 x + 24 
= Smith's age 3 years ago. 

15 X— (x-1-6) = 14x— 6=the difference between their ages 3 years hence (1) 
(5x+24)— x=4x+24 = the difference 3 years ago (2) 

But since the difference always remains the same, we may put the 
expressions (1) and (2) equal to each other, thus: 14x — 6 = 4x + 24, 
whence x = 3. Therefore John is now 3 + 3 = 6 years, and his father 
42 years, old. 

196. Let X = time the second hand is equally distant from the other 

X 

two. X — 60 = distance second hand is past 12. — = distance minute 

^ 60 

hand is past 12. — = distance hour hand is 

^ 720 

past 12. Now, the distance the second hand 

is past 12 = J of the sum of the distances the 

hour and minute hands are past 12. Hence 

2(x - 60) = — + — , or 1440X ~ 86400 = 12x + x, 
60 720 

whence x = 60 ^'//^ seconds. Similarly, we obtain 

^ = X - 60 + -^, whence x = 61 M^, the time 
60 720 ''^' 

the minute hand is midway between the others. Finally, in like manner 




ALGEBBAIC SOLUTIONS. 191 

we e:et (60 — x)-\ — - = — , whence x = 59H seconds, the time till the 

hour hand is equidistant from the other two. (From Dr. I. J. Wirbback.) 

197. When C reaches IV, A will be at Z>, 5 miles from R. Let U be 
the point where C picks up A. Then, since C travels 8 times as fast as A, 
DE = ^ of DW=S^g, Let UG = the distance C carries A, and x = GW. 
Then EG = So - S^ - x = 31i - a:. A walks 5 + 3| + x. Therefore the 
time of the trip in hours is 

5 + 3| + x + K3H-a^) (1) 

Let RF be the distance traversed by B until C leaves A at (r. As B's 

rate is 2 miles an hour, 

i?i^ = 2 f 5 + 3| + ^^\~ ^ ] = 25 J 



FG = 4.0-(25i-lx)-x=U^--lx. 

Let H be the point where C picks up B. C's rate : B's : : 4 : 1. Hence 
FH=l(U^-ix) = 2^j-^%x, which B walks. IIG = FG-FII=ni-%x. 
HG + (^PF= 11| + f a:, the distance C carries B. 

Note. — We know C carries B to W, because, if he put him down at any point between 
JJand W, C would reach ^first, which is not allowed by the condition of the problem. 

While A walks from G to TF, B walks from F to H^ and is carried from 
J? to W. Hence, as J the distance he walks and ^ the distance C carries 
him = the time, we have J(2| — 2^0 ^)+ 8(1^1 + l^)= ^^^ Hvhq from F 
to TF, which must be x hours, as in that time A walks from G to W, 
Letting this expression for the time = x^ and solving, we find x = 2|§|, 
Substituting this value of x in (1), we find the time to be 15^^ hours. 



RADICALS. 



198. (a - x) \h-^— = A --^— (a - xy = Va^ - x-. 

Then (a — x) Va^ — x^ — Va^ — x^ = (a — x — 1) Va'-^ — x^. 

199. (i_a;2)-i + i3.l±vrH^. 

Vl-X2 

The numerator may be written 



1(1 _ ^)_|- 1(1 4. x) + vT^^^. 



192 TABLE BOOK AND TEST PBOBLEMS. 

Hence the fraction becomes 





Ki- 


x) + Vl-x'' + 


K1 + 


X) 


Taking the 


square root, 


we obtain 








Vki 




-J, 





the root required. 

we have 

/''V _L /y^ 0.-\//y^' ■\/'V« -x/n 



200. Squaring, we have 

(x + g)— 2-yyax _ Vx — Va 
^ — (^ Vx + Va 

Multiplying both terms of the right member by Vx + Va, we have 

{x-\-d)—2 -y/ax _ x — a 



- (2) 



Now, X — a being a mean proportional between (x-\-a) — 2 Vax and 
(x + a) +2 Vax, the equation is true in the forms (1) and (2), and also 
in the original form, as the ratio was preserved all through. 



and 



201. This may be written iiL + 1^ Vm'^ - n^. 
^ 4 2 

- Vm2^^^ = 2f- Vm2 - ?z2 V 
2 U / 

•2\ 



4 2V 2 



but f-Y+( •^^^ - W^ V := «! 

Uy V 2 j 4" 



Hence f - \V - Vm'^ - n'^ + 

2i j 2i 



Vm'^ - n'^V 



= the quantity whose root is to be found, and - H is the root. 

Z 2i 



QUADRATIC EQUATIONS. 

202. Let X — the height of stump. Then a — x — part crossing stream, 
(a - x)2 = &2 ^ ^2^ or a2 - 2 ax + x2 = &2 + a;2, whence x = ^\ ^ » This 

formula furnishes the following rule for finding the height of any stump 
when the conditions are similar : From the square of the height of the 
tree subtract the square of the stream's width, and divide the remainder 
by twice the height of the tree. 



ALGEBRAIC SOLUTIONS. 193 

Let d = the diameter. Then d^ x ^tt = — = contents of sphere, 

idV ^ 

and l-j X 4 TT = TTcZ^ = surface. By the condition of the problem, 

trd^ = ^, or 6 ird^ = ird^. Dividing by ird'^y we have 6 = (^, the diameter. 

204. First Solution, 

From (2), ?/ = 5. 

X 

Substituting in(l), x + -=:a. 

X 

Clearing, we have x^ — ax = — 5. 

Completing square, etc., x = - ± - Va^ — 4 6. 

Second Solution, 

From (1), x = a — y. 

Then (2) may be written 

or y'^ — ay = — b, 

whence 2/ = - ± - Va^ — 45 . 

205. (2)-(l)2 = 2v^ = 20-4a;2/. 
Transposing, reducing, etc. , 

xy -h i Vxy = 5. 



Completing square, etc., 


^/xy = 2 and — |. 


Whence 


xy = 4: and -2^. 


But 


x = 20-y. 


Hence (20 - 


-y)y = 4: or -2^. 


Solving this, we find 


?/ = 10 ± 4 V6 and 10 di f VT5. 


Also 


X = 10 ± 4 VG and 10 ± f vTS. 


206. Let 


a:- = the number. 



Then, by conditions, 

Vx4-6 - Vx-G = 6, 
or Vx + 6 = 6 + Vx -6. 

Squaring, a: + 6 =r 36 + 12 VaT-^ + x - 6, 

or 12 \/x-6 = - 24, 



and Vx-e = - 2. (1) 

ellwood's test prob. — 13. 



194 TABLE BOOK AND TEST PROBLEMS. 

Squaring again, we have a: — 6 = 4, 

or X = 10. 

To verify, put 10 for x in the first equation, thus : 

VTO - V4 = 6 (2) 

Extracting the indicated roots, 

4-(±2)zz:6. 

Hence (2) is true v^^hen Vi = — 2, as it does in this problem. [See 
Equation (1).] 



207. Adding 2, 


x2 + 2 + l = 2. 

X2 


Eactoring, 


(-!)"-• 


Extracting square root, 


X + - = ± V2. 

X 


Clearing, etc., 


x2±xV2= - 1. 


Completing square, x^ 


±x\/2 + J= -J. 



Whence x = ± Vj db V— J, 

which may be written — — — — — — 

V2 

208. Let X = number of gallons first drawn. 
Then 81 — x = remainder. 

— = the part drawn, 
81 ^ 

and —(81 — x) = number of gallons wine drawn 

^^ 2d time. 

81~x-— (81 -x)=36. 

8r ^ 

Clearing, 81(81 - x) - x(81 - x) = 81 x 36, 
or (81 - x) (81 - x) = 81 X 36. 

Extracting square root, 81 — x = 9x6 = 54, 

whence x = 81 — 64 = 27. 

Then ||(81 - 27)= 18, number of gallons drawn 2d 

time. 

209. Let X = the width of frame. 

Then 10 + 2x = the width of picture and frame, 

and 16 -I- 2x = the length of picture and frame. 



ALGEBRAIC SOLUTIONS. 



195 



Then (10 -f 2x) (16 + 2x) = area of picture and frame. 

By the problem, (10 -\- 2x)(16 -\-2x)= 320, or 4^2 + 52x = 160, 
and x2 + 13 X + ^F = 40 + ^F = -|-- 

Extracting square root, x + -^2^- = ±i V329 = 9.069 + . 

.-. X = 9.069+ - 6.5 = 2.569+ inches. 
X = length of a side in rods. 
x^ = the area, 
4ic = its perimeter. 



210. Let 

Then 
and 
By the problem, 

Clearing, 
and 
The area is 



4x=:-^. 
160 

640 a; = x% 
X = 640. 

— = 2560 acres. 
160 



211. The first member may be written thus: 



1 + 



2\/n 



— ; and the second member, 1 H -- 

ax — Vn Vn 

2Vn Va 



^ — Vn -\-2Vn 
Vox —Vn 



or 



Hence 
Clearing, 



'ax — Vn Vn 

2n = aVx—y/an^ or aVx—yfan — 2w. 
y/an — 2n 



Vx = 



Squaring, we have 



an — ^n^/an + 4yi2 _ n{a — 4:Van + 4n) 



a' 



_ n(Va-2Vny ^ 
a 



212. Completing square, 

Extracting square root, x - ^~^ + ^ = -^(6 - a)c + [ ^ ~ ^ + ^ V^ (1) 
Reducing the quantity under the radical sign, it becomes 



^ a^-2ab - 2ac + b^-h 2bc + d^ 

\ 4- ' 

and factoring, it becomes 

JCcT-by- 2c(a - b) -h c^ 



196 TABLE BOOK AND TEST PROBLEMS. 

Substituting in (1), we have 

a-b + c _ f(a-6)^- 2c(a - &) + ^ 
2 > 4 

Extracting square root of right member, 

^ c(' — f>-\-c _ , a —b—G 

JO — — I— • 

2 2 

Whence x = a — 5, or + c. 

213. Let X = distance army traveled before the officer turned. Then 
25 + x = distance the officer travels in same time. 25 — cc = distance yet 
to be traveled by the army. 25 — (25 — x)= x = distance officer travels 
in returning. Then, since the rates of travel remain the same, the dis- 
tances are proportional, as follows : a: : 25 + a: : : 25 — x : x, whence x = 
17.68 + miles, and 25 + 2aj = 60.36 + miles, the distance traveled by the 
officer. 

214. Let X = the distance from the louder bell, and a — x = the distance 

1 0*2 

from the other. Then by condition, 1 : 3 : : x^ : (a — x)'-^, or - = 



3 {a- x)2 
\ j_ X a 
Extracting square root, — :: = -^ — , whence x = a — x = a — 

^S a-x 1±V3 

a _ ibav3 

1 ± V3 ~ 1 ± \/3 

215. Let x = distance from B, the stronger light. Then 132 — x = 
distance from A. By condition, we have 7 : x^ : : 17 : (132 — a^)2, or 

— = Extracting square root, = ± Clearing, etc., 

^2 (132 — x)'^ x 132 — x 

X V7 ± X VT7 ^ 132 V7. Whence x = ^^^^^_ . 

V7 ± Vl7 

12 144 

216. Let X = number of eggs for 12 cents. Then — x 12 = = price 

X X 

per dozen ; also x 12 = = decreased price per dozen. Then, 

X + 2 X + 2 

144 144 

by problem, = 1, or x"^ + 2x = 288 ; whence x = 16 eggs, the 

X X + 2 

144 144 

number given for 12 cents. « — = — = 9 cents per dozen. 

X 16 



217. Squaring, 1+^ = 1-^ + 2\/l - - -I- 1. 

a X ^ X 

Transposing, 2\/l — = -H 1. 

® ^ X a X 

Squaring, 4 -- = (^ + -) - 2(- + -) + 1 (1) 



ALGEBRAIC SOLUTIONS, 197 



and addino; — • 



Now, taking 4 


from ( 


' X 

- + 


9' 


we 


have 


{I- 


ay 


to — 2 1 - + - ) , we have 
\a xj 


-2 


e- 


-1} 








Hence (1) may 


be written 












e- 


■r- 




■:) 


+ 1: 


= 0. 






Extracting square root, 


X. 

a 


a 

X 


-1 


= 0, 






or 






X^- 


- ax 


= a\ 






Whence 








X 


_ a _^ ^. 


V5 


= ''(1 



218. Let X = cost of husking a standing row, and tj = that of a " down " 
row. Then S x : x -\- y : : 2 x -{- y : S x, whence 1 x^ = Sxy + |/"^, (1). 
y\ being " down " rows, 2y -\- 9x = 20, whence y = 10 — 4 J x. Putting 
this value in (1), we find a- = $1,655+; and 9x, the standing rows, 
= $14.90. Then $20 - $ 14.90 = $5.10, cost of " down " rows. bx-]-y 
= A's money = $10.83, and 4z + ?/ = B's money = $9.17. 

219. This equation may be written 

^4a _ 2 x3« + X2<» — x2« -}- X^ = 6. 

Factoring, (x^^ — x'^y — (x^^ — x«) = 6. 

[From this quadratic we find 

^2a _ ^a _ J _jL VoJ = 3 or - 2. 

Taking x^^ - x« = 3, 

we find x^ = l ±1 Vl3 = K^ ± ^1^)- 



Hence x = Aq(l±Vl3) . 

Taking x2« - x« == - 2, 



we find x" = ^ lb ^? V 



Hence x='\/i(l ± V- 7). 

220. Let X = the digit in tens' place, y = that in units' place. Then 
lOx -\- y = the number. 
By the first condition, 

10x2 4- ^2/ = 46 (1) 

By the second condition, 

x{x + ?/) or x2 + xy = 10 (2) 



198 TABLE BOOK AND TEST PROBLEMS, 

(l)_(2)gives 9x2 = 36, 

whence x = 2. 

Then from (2) we find y = ^^ 

and we have 10 a: -f 2/ = 20 + 3 = 23, the required number. 

221. Let X = one, and y = the other. 
Then x + y = xy = x^ — y^, 

from which we have 

X2 - ?/2 rr: (X + ^/) (x - tj) = X -\- y. 

Dividing by (x + ?/) , we get 

x-y=l, 
whence x = ?/ + 1, and y = x — 1 (1) 

Then xy = y(y -\- 1)= y'^ + y = x -\- y, 

whence y^ = x, then y = Vx (2) 

Equating values of y as found in (1) and (2), we have x — 1 = Vx. 

Squaring, etc., x^ — 3 x = — 1. 

Adding I, x^~Sx-\- 1 = 1 

whence x = ^ + J VS. 

Then y = x - 1 = }y -^ h \/l = i{l -}■ VI), 

222. Clearing, 



Va — V a — Va^ — ax = nV~a + n\ a — Va- 



ax. 



ax. 



Collecting, Va(l - rO = (^^ + 1) v a - Vo^ 

Squaring, a(l — 2 ?i + n-) — {ii^ + 2n + l)(a — Va^ — ax), 

or a — 2 an + an^ _ q^^i^ j^2an -{- a —{ji + 1)2 Va- — ax. 
Collecting, —Aan — — {^n-\- 1)2 Va- — ax, 

or _^^'L_^V5^:rirx. 

(« + 1)^ 

Squaring, -^^ ^— - = a2 — ax, 

^ 0^ + 1)* 

a2(2V7i)4 o 

or — ^ — = a^ — ax. 

(n + 1)* 

Transposing, ax — a- — a- — ] ? 

and x = a — al — ] = al I — I ) ) • 



ALGEBRAIC SOLUTIONS. 



199 



223. Let a, b, and c — the numbers. Then =z - 

2ac b a c 



1 



, whence b = 



a -{- c 
By first condition, 



a + A^ + c = 191 



a + c 

By second condition, ac = 4032 

From (1), a^ -f 4 ac + c2 = 191 (a + c) 

Subtracting 2ac ■= 8064 from (3) , we have 

(a + c)2 = 191 (a -f c) - 8064. 
Transposing, and completing the squares, 

^a + c)2 - 191 (a + c)+ SKs^ = 9120.25 - 8064 = 1056.25, 
whence a + c = 128, and c = 128 — a. 

Then (2) becomes ac = a (128 - a) = 4032, 

or a2 _ 128 a = - 4032. 

Adding 4096, a' 

whence 



.2 _ 128 a + 4096 = 64, 

a = 72 or 56. 



Then 



c = 56 or 72, and b = 



2ac 



(1) 

(2) 
(3) 



= 63. 



224. Let 2x = 

and X = 

Then 2 x - 6 = 

x-6 = 
and a; — 3 = 

2^3 = 
(2a;-6)(x-6)(:^-3) = 

The difference between the solidity of 
tion must be, by the last condition, 11,7 

Hence 2x^-{(2x-6)(x-6)(x-S)}=: 
or 2 x3 - (2 x3 - 24 x2 + 90 X - 108) = 
or 

whence 
and 

Then the trough is 



and 



4x2 -15x 

X 

2x 
48-6 
24-6 
24-3 



: length of log, 
: width and depth. 
: length of excavation, 
: width of excavation, 
: depth of excavation, 
: solidity of stick, 
: solidity of excavation. 

the stick and that of the excava- 

72. 

: 11,772, 

: 11,772, 

: 1944, 

: 24, the width and depth of stick, 

: 48, the length of stick. 

: 42 inches long, 

: 18 inches wide, 

: 21 inches deep. 

(From Dr. I. J. WiREBACK.) 



200 TABLE BOOK AND TEST PROBLEMS. 



225. Let X = the first, y 
conditions, we have 


= the second, and z 

y(x + z)=bO 
z(y-hx')=m 


= the 


third. 


Then 


, by 

(1) 
(2) 
(8) 


By(2)-(i), • 


z(y — x')= 24: 








(4) 


By (3) + (4), 


yz =40 








(5) 


By (3) -(4), 


0:^ = 16 








(6) 


By substituting the value 


of xz in (1), 

xy = 10 








(7) 


Erom (5), 


_40 








(8) 



Erom 


(5), 


From 


(6),- 


Hence 


J 


whence 
or 




From 


(7), 


Therefore 


From 


(10), 



y 

16 

z = — 

X 

16 ^40 

X y' 

16y = 40 X, 
x = iy (9) 

x='^ (10) 

2 V 10 

-^ = — , or 2 2/2 =r 50, and y-h, 

x= — =2, and from (6), ^ =: — = 8. 
y ' ^ ^' X 

226. This solution assumes that B receives 3s. 9d. less than he would 
have received had C not been called in, in which case they would have 
worked longer than 5 days, and A would have received more than 60 
shillings. For another interpretation of the italicized clause, see "Arith- 
metical Solutions." Let x = the time required by B, and y = C's time. 

90 
A should have 10 shillings per day, and B — shillings per day. In 1 

X 

day A does -, and B 1, of the entire work. Then of- + iA = ^(^ -^ ^^ = 
^ 9' X V^ x) 9x 

q /v 
part of the whole work A and B can do in 5 days. — ^-^ — = time in 

a: + 9 

which A and B together can reap the field. Then — x — — = = 

90 450 

B's pay if C had not been called in. — x 5 = — = what B did receive. 



ALGEBRAIC SOLUTIONS. 201 

Therefore = 3f , or — = -, whence x^ — SI x = 

X + 9 X X -\-9 X 4 

— 1080, and x = 15 days, B's time. As the work was completed in 5 

5 5 2 5 5 2 

days, we have -H f--r=l, or -H f-- = l, whence y = IS days, 

9 X y 9 lo y 

Cs time. (From Mathematical Magazine.) 

227. Let X = the length, and 6y = the original rate. Then 6y x 1 — 
by = distance traveled before accident, and x — 6y = distance yet to run. 

^-^ — ^ = regular time of running this distance. Hence - ( ^-^^ — ^ 
by S\ by 

increased time of running this distance = ^ hours. Then ^ 

Sy sy 

4- 2 = the time on whole line, which is 3 hours more than regular 

X 

time, ^. Hence the equation 
by 

y- - 5 ?/ ^ = A + 3 (1) 

S-y ^ by^ • ^^ 

Under the second condition, x —(by + 50)= distance to run after acci- 
dent, and the time to run it is ~ '^ J ~ ^ hours. -^ — = time on 

Sy by 

road before accident. Hence ^ 1 — — --^-^^ h 1 = whole time 

5^ Sy 

on trip, which is 1^ hours less than under first condition. Hence the 

equation 

by -]- bO ^ X - by - bO , ^ _ f X , ^\ -, i ' .„. 

From (1) we find x-20y = (3) 

From (2) we find x-lOy = bO (4) 

Subtracting, 10?/ = 50 ; y = b ; by = 25, the original rate. From (3), 
X — 20y = X— 100 = 0; and x = 100, the length of road. 

228. Let X = the first number, and y = the ratio. Then x^ xy, xy''^, and 
xy^ will be the numbers. 

X -{■ xy -\- xy'^ + xy^ = 15 (1) 

and a:2 + xY" + ^^V^ + ^'^2/^ = 85 (2) 

Factoring (1), we have x(\ 4-2/)(l + y'^) = 15, 

whence x^ = -^ — ■• (3) 

(1 + yyii + yy 



202 TABLE BOOK AND TEST PROBLEMS. 

Factoring (2), we have 

whence x^ = — (i) 

Equatmg (3) and (4) , and dividing by , we have 

1 + if 

45 17 



(1 + 2/)'^(i + y') 1 + 2/' 

Clearing, etc., 45 + 45 ?/4 = 17 + 34 ?/ + 34 2/2 + 34 a/^ + 17 2/*, 
or 28?/4 + 28-34?/3-34?/ = 34 2/2. 

Dividing by y^, and factoring, we obtain 

28(2'^+^)-34(2/+^)-34,orl4(,^ + ^^)-17(2/ + l)=17 (5) 

Put y-i--z=m. Then ?/2 -j- _1 = wi2 _ 2. Then (5) becomes 14 m^ - 
2/ 2/^ 

17 m = 45, whence m = J| =: 2i-. Therefore y -\- - = 2}. Clearing, etc., 

2/ 
2/2 — 1 2/ = — 1. Completing the square, ?/2 — |^ -|- i| = -9_^ whence 2/ — f 

= dbf, or ?/ = 2. Therefore x = 1 ; and the numbers are 1, 2, 4, and 8. 

229. Let X = one number, 

and y = the other. 

Then xy=p (1) 

and x^ — 2/"^ =r m(x — ijY (2) 

Dividing (2) 'by x — y, we have 

^2 + X?/ + ?/2 rr m(X — ?/)2 (3) 

Treating (3) as a proportion, we get 

^2 + a*?/ 4- 2/^ : (ic — 2/)'-^ : : m : 1. 
By division, 3 x?/ : (a: — yY : : m — 1 : 1. 

Multiplying the first term of each ratio by |, and clearing, 

4iXy : (x — yY : : 4 m — 4 : 3. 
By composition, 

(x + ?/)2 : {x-yY :: 4m- 1 : 3. 
Extracting square root, 



X -\- y : X — y : : ^4 m ~ I : VS. 



ALGEBBAIC SOLUTIONS. 203 

By composition and division, 

X : y : : V4 m — 1 -\- VS : \/4 m — I — v^3. 
Multiplying the first ratio by y, 



xy : y'^ '' : V-i m — 1 + VS : V-i m — 1 — v 3. 
But a:?/ = p, 

hence ./ ^^p(V^^_V3) ^ jX VI^JT^l - VS)^ , 

V4 7)z-1+V8 4m -4 

Then ,^^l/y K4m-l)-V3]^ N 

and from (1), x = I PV^^^^ 

\/p(4 7H — 1) — VSp 



/Second Solution, 








Let 




X — one, 2/ the other ; 




then 




xy^p 


(1) 


and 


x^- 


- if = m(x — ijY 


(2) 



Dividing (2) by x — ?/, and subtracting 3p, 

(x — yY = wi(x — 2/)^ — 3^, 
or (m — 1) (x — yY = Sp ; 



whence x = y±\ — ^ — 

\m — 1 

Then (1) becomes 

\ m — 1 



whence y = 1 / db Vp(4m - 1)± vgj\ . 



204 TABLE BOOK AND TEST PROBLEMS, 



SOLUTIONS TO SPECIAL EXPEDIENTS, 
SIMULTANEOUS EQUATIONS. 

230. First Solution. 

Adding (1) and (2), x'^ -h x -\- y'^ -j- y = 18. 
Adding J to both members, 

x^ + x + i-^if-^y-\-l = lSi = -V- = -V- + -y. 
By inspection we know x'> y. 
Hence x^ + x + i = V-, 

and i/^y -{- 1= 2^5. 

Whence x = S and — 4, and y = 2 and — 3. (This solution is faulty.) 

Second Solution, 

From (1), x2 + ?/ =: 9 + 2, 

or x2 - 9 = 2 - ?/ (3) 

From (2), x + y'^ = i + S, 

or X - 3 =: 4 - 2/-^ =(2 - y)(2 + ?/), 

or '^ = 2-y (4) 

2 + 1/ 

Equating (3) and (4), ^2 - 9 = ^^^, 

or x^ — 9 



2 + 2/ 2 + i/ 



:« n 3 



Transposing, x'^ ^ — = 9 — 

^ ^ 2 + 2/ 2 + 2/ 

Completing square, 



2 + 2/ 4(2 + 2/)--^ 2 + 2/ 4(2 + 2/)^ 

Extracting square root, x — = 3 — 



2(2 + 2/) 2(2 + 2/) 

whence x = 3. 



Substituting in (1), we find y = 



— 9 



SOLUTIONS TO SPECIAL EXPEDIENTS. 205 



Third Solution. Make 


2 +?/ = «. 






Then from (2) we have 


x-3 = s(2-2/), 






or 


X 3 ^ 

s s -^ 




(3) 


From (1), 


x^ -9 = 2 -y 

•2 o »^ 3 

/. x^ — 9 = , 

s s 




(4) 


or 


s s 




(5) 


Adding — to each member, we have 

^■2 ^+ 1 =9 §+1. 






Extracting square root, x = ±(3 h 

2s V 2s/ 

whence one value of x is found to be 3. 






Fourth Solution. 


x^-\-y = n 




(1) 




X + 1/2^ 7 




(2) 


From (1), 


y = n-x\ 






Then 


?/^ ==121 -22x2 -f 


X* 


(3) 


From (2) 


if = 7--x 




(4) 


Equating (3) and (4), we 


have 






121 


-22x24-x4 = 7 -X, 


• 




or X* - 22 


x2 + X + 114 = 




(5) 


Factoring (5) , we have 








(x-3)(x3 + 3x2 


-13x-38)=0. 






Placing each factor equal to 0, the values of x are readily found. 




Fifth Solution, 


x2 + 2/ = 7 




(1) 



X + 2/2 - 11 (2) 
From (1) and (2), by transposition, we get 

y2_g = 2-x (3) 

and 2/ - 3 = 4 - x2 (4) 

Put 2 + X = W, and 2 — x = m. 

Then (3) and (4) become y^ - 9 = m (5) 

y-S = mW (6) 



206 



TABLE BOOK AND TEST PROBLEMS, 



Equating values of m as derived from (5) and (6) , we have 

..o r. tf-S 







' ''- w 




Therefore 




Wy'^ ~ij = 9W- 3 


(7) 


or 




y^ y=^-^ 
^ w w 




Completing square, y'^ 


— 


yv 4:W^ w 4Tr2 




Extracting square root, 




2/ ^=3 V. 




Therefore 




2/ = 3, 




and 




x = 2. 








(From Professor B. 


F. BUKLBSON.) 


Sixth Solution. 












x2 4- 2/ = 11 


(1) 






x-{-i/ = 7 


(2) 


(l)x 


'^y 


= x'^y -\-if = lly 


(3) 


(3)- 


(2) 


= x^y — x=lly — 7 


(4) 



or 



(4) 4- twice (1) = x-?/ + 2 x2 - X = 9 2^ + 15, 
x^(y 4- 2)- a: = 9?/ + 15. 



Dividing by 2/ + 2, and completing the square, 

^2 X 1 ^ 36?/2 4-132?/4-121 

I/ + 2 4(2/ + 2)2 4(2/ + 2)2 

Extracting square root, we have 



X — 



1 



- ± ^/ + 1 1 



2(2/ + 2) 2(2/ + 2) 



or 


x=^y-^'^ 

2(2/ + 2) 


whence 


x = S. 

(From Dr. I. J. Wireback.) 


Seventh Solution. 






y^-\-x = n (1) 




x2 + 2/ = 7 (2) 


From (1), 


y^=n-x (3) 


From (2), 


2/2 = 49- 14x2 + x4 (4) 



SOLUTIONS TO SPECIAL EXPEDIENTS. 207 

Equating and transposing, 

X*- 14x24-x= -38. 
Adding to each member 2 x^ + 4 x'-^ — 20 x, we get 

x4 + 2 x3 - 10 x2 - 1 9 X = 2 x3 + 4 x2 - 20 X - 38. 
Factoring, x(x3 + 2x2 - lOx - 19) = 2(x3 + 2 x2 - 10 x - 19), 
whence x = 2. 

Therefore y = S. 

231. By(l) + (2) + (3), 

2(r2 + s^ + ^■^) + rs -\- st + rt = m'^ + n^ + p^ (4) 

By (4)2, 

4(r2 + s2+^2)24.4(^2_|_52_|_^2)(^5_|.Sj^,.^) + (^5 + S^_j_^^)2_(^2_|_^2_|.p2)2 (5) 

By 2(1)2 _|_ 2(2)2 ^ 2(3)2, ^e have, by factoring, 



By V(5)-(6), 
1-5 ^_ s^ _l_^t = -i- V{i[(2 ?7i2,^2 _|_ 2 ri2p2 _^ 2 m2j)2) -.(^rii^ + n* + p*)]} (7) 

Put right member = c. 
Then, by (6(7) +2(4)}*, 

2(r + s 4- = ± V{2(??i2 + ^2 +^2) ^Qc} (8) 

By (4) + (7) - 2(2), and factoring, 

2 r(r + s + = ^^^ - ^^ +i^^ + c (9) 

, By (9)^(8), 

r = (m2- n2+j92_^ c) -^ ± V{2(m^ + ?i2 + p^) ^ g g}. 
Similarly we find 



S =(n2 -p2 _|_ 9,^2 _|_ c)-f-±\/{2(77l2 + 7^2 4. ^2) 4. 6 c}, 

and t = (p^- m2 + ?i'2 -f c) -^ ± V{2 (^2 + 9i2 + ^92) + 6 c}. 

(From Professor B. F. Burleson.) 



5. From (1), x = 10 - y. 

Then (2) becomes (10 - y) Vy = 12, 

or .10V^-?/\/?/ = 12 (3) 

Put \/y z= m. 



208 TABLE BOOK AND TEST PROBLEMS. 

Then (3) becomes 10 m — m^ = 12, 

or m3-10m+12=:0 (4) 

(4) X m = m* - 10 m^ + 12 m = 0, 

or m^ - 6 m2 + 9 = 4 m^ - 12 i?i + 9. 

m'^ = 2m. 
m — 2 — V?/. 
Then 2/ == 4, 

and X = 6. 

233. Adding twice (2) to (1), 

x2 + 2 a:?/ 4- ^2 = 64. 
Extracting square root, x -\- y — ±^ (3) 

Subtracting twice (2) from (1), 

x^ — 2 a:?/ + ?/2 = 4. 

x-y=z±2 (4) 

Comparing (3) and (4) , we find x — b, and y = Z. 

234. (2)-(l)z=2(2/-x) = (?/-x)(2/ + x), 
or 2 =: ?/ 4- :r. 

Then (1) becomes x^ = 2, 

X = di V2, 
?/ = 2 d= V2. 

235. Firs^ Solution. (1)2 = x'^ + ^/^ = 16 - 2 x?/ (3) 

(3)2 = X* + 2 x22/2 4- 2/i = 256 - 64 x?/ + 4 x2?/2, 
or X* - 2 xhf + 64 x?/ + ?/^ = 256 (4) 

X* 4- 2/* = 82 (2) 

Subtracting, 64 x?/ — 2 x2?/2 = 174, 

32 xy - xhf- = 87, 
or xY - 32 x?/ 4- 256 = 256 - 87 = 169. 

xy-16 = ± 13. 

xy = S or 29 (5) 

From (5) and (1), x is found to be 3 or 1 ; and ?/, 1 or 3. 

Second Solution. 

(1)4 := X* 4- 4 xhj 4- 6 x2?/2 + 4 x?/3 + ?/ r= 256 (a) 

(a) + (2)= 2x4 4-4x^2/ 4- 6x2?/2 + 4txy^ + 27/ = 338, 
or X* 4- 2 x'^y + 3 x2?/2 ^2xy^-\-y^ = 169. 



xy(x- 


-y)-- 


= 6. 






X 


-y = 

xy = 


= 1. 
^6 








X = 


= 2/ 


+ 1. 




{y + 


1)2/ = 


r6. 








2/ = 


= 2 


and 


^3. 




X - 


= 3 


or - 


2. 



SOLUTIONS TO SPECIAL EXPEDIENTS. 209 

Extracting square root, x'^ + xy + y^= ±1S (b) 

(1)'-^ = X'^-\-2 Xy + 2/2 zr: 16 (c) 

(c) - (&) = X2/ = 3 or 29. 
Hence cc = 3 or 1, 

y = 1 or 3. 

236. (2)-(l)^ 
But 

Hence x?/ = 6 (3) 

From (1), 

Then (3) becomes 

Completing square, etc., 

Then from (1), 

237. Putting x + ?/ = s, and xy = j9, the equations become 

s+ps-{-p^ = S5 (1) 

and p -{- s^-\-ps=:97 (2) 

Adding (1) and (2), (p2 + 2ps -\- s2) + (p + s) = 182. 

Completing square, (p + s)^ + (p 4- s) + i = -|- ; 
•whence j? + s -f J = Y-, and p + s = 1S, > 

or j9 = 13 — s. 

Putting this value of j9 in (2), we find s = 7. 

Hence p = 6, 

Then x-{-y = 7, 

and xy = 6 ] 

whence x = 6 or 1, 

and y = 1 or 6. 

238. Squaring (1), -^ + ^-±-^ = 2. 

X + 2/ 3x 

Putting X -\- y = s, and clearing, we have 

9 x2 — 6 X5 = — s2. 

Completing square, 9 x^ — 6 xs + s^ = ; 

whence 3 x— s = 0, and 3 x = s. 

Therefore x + y = ox, 

or ?/ = 2 X. 

Substituting in (2), 2 x2 - 3 x = 54 ; 

whence x = 6 or — 4|, 

and 2/ = 12 or — 9. 

14. 



210 



TABLE BOOK AND TEST PROBLEMS. 



239. First Solution. 
(l)-(2) = 

whence 

and 

From (2), 



xy — y'^ = 15 



y 



x-^ 



_ 225 + 30^24^ 



y 



x2 = 89 - 2/2 



(3) 
(4) 



Equating (3) and (4), and clearing, 

225 + 30 2/2 + ?/* = 89 y^ - 2/4, 

or 2/* - ¥ 2/^ = - -f-- 

Completing square, y^ - -s/- 2/^ + (-\9)2 - (5_9)2 

41 



225. 

2 



1681 
16 • 



'2 - \^- = ± 4-1- 



whence 
Therefore 

Second Solution. 

Let 

Then (1) and (2) become 

whence 



whence 



y 



5 or SVh 



(From Siipt. S. Transeau, in Educational News.) 

x2 + 2/2 = 89 
x"^ -{- xy = 104 

X = vy. 

^2y2 4- 2/2 rz 89 



2/2 = 



^^2+1 



Equating, 
Clearing, 



v^if + vy^ = 104 
2 104 

89 104 



(1) 
(2) 

(3) 
(4) 



or 
or 



104 v2 + 104 = 89 i?2 _f 
lbv^-Sdv= - 104, 



V, 



9 75 — — -IJti 



13 



Completing square, etc. , we find 

Substituting in (3), we find y = ± 3VJ or ±5. 

Then x= ±yV2 or ± 8. 

(From J. M. Peoples, in ^Educational News.) 



SOLUTIONS TO SPECIAL EXPEDIENTS. 211 

240. Let x = 771 -\- n, 

y = m — n. 
Then (1) and (2) become 2 m^ - 2 mn'^ = 70 (3) 

and 2 m3 + 6 mn^ = 133 (4) 



Eliminating mn^^ we find 


m^^^l^ 


or 


m = J. 


Substituting in (3) , we get 


w = |. 


Hence 


X = m + w =: J + f = 5, 


and 


2/ = m — n = J — f = 2. 



241. First Solution. 

(1)3 = x3 + 3 x^?/ + 3 xy^ -hy^ = 125 (3) 

Subtracting (2) from (3) , Sx'^y + S xif = 60, 
or xy(x + y)= 20. 

But ' X -i- y = D. 

Hence 5 x?/ = 20, x?/ = 4 (4) 

Erom (1) and (4), y is found to be 4 or 1, and 

x= I or 4. 

Second Solution. Dividing (2) by (1), we obtain 

■x'^-xy-{-if = lS (3) 

Squaring (1), x'^ -j- 2 xy -{- if = 25 (4) 

(4) -(3)- Sxy=:12, 

xy = 4:. 
X and y may now be found as above. 



242. Let 




X = vy. 






Then (1) becomes 




^2^2 ^ ij2 = 34^ and y^ . 


34 

^2 + 1 


(3) 


(2) becomes 




^2y2 _ ^.y'2 ^ 10, and 2/2 : 


10 


(4) 


Equating (3) and (4), 




34 10 

^2 _|_ 1 ^2 _^ .^ 






Clearing, completing, 


etc.. 


we find -^ = 11 or —J. 






Substituting in (3), 




¥2/^ + 2/^=^=34, 







whence ?/ = 3. 

Then x = 6. 



212 



TABLE BOOK AND TEST PROBLEMS. 



243. Let 

and 

Then 
and 

Put 

Then (1) becomes 
whence 
or 

Then 

From (2), putting the value of v equal to a, we have 

(f.y2 _^ yl — gZy^ _ yZ^ 

or a*2 + 1 = ahj — y, 

/72 J- 1 



X 


= one 


number, 


y 


= the other. 


xy 


= X^- 


.yz 


X2 + 2/2 


= x^- 


.yZ 


X : 


= vy. 




V2f 


= v'^f 


-y^\ 


V ■ 


:=V2_ 


1. 


v^~v 


= 1. 




V 


= i^ 


\y/l = \ 



(1) 

(2) 



(1 + V5). 



*T i.i.\j i.i.\J\J 






a^ — 1 




Substituting the value of 


cc> iO- + V5), we have 










1+V5 ^ 




Then 






a^ = i(5+V6). 




244. Let 




X = one number, 




and 






y = the other. 




Then 




(X 


-?/)(x2-2/"')=32 


(1) 


and 




(X 


+ y)(x'^-\-y^)=272 


(2) 


Put 






x-{- y = s, 




and 






xy=p. 




Then 


(1) becomes 




53 _ 4ps = 32, 




and (2) 


becomes 




s3 _ 2ps = 272, 




whence 






ps = 120. 




Then 






s3 - 2ps = s^- 240 = 272, 




or 






s^ = 512, 




and 






s = S = x-\- y 


(3) 


Also 






p = i|o = 15 =r xy. 




Prom 


(3), 




x=:S--y. 




Then 






xy = y{S-y)=:15, 




whence 






y = 6. 




Then 






X:=S. 





SOLUTIONS TO SPECIAL EXPEDIENTS. 213 

i. Let xy = p. 

Substituting the value oi x -{- y from (2), andp for xy, (1) becomes 

55:=(9-2i))(27-8i9). 
Developing, completing square, etc., we have 

i) = f I or 2. 
Hence -f-f or 2 = xy (3) 

From (2) and (3), we find x = 2, and y = 1. 



246. Let 




x^ =r, 

y^ = s. 




Then 




r-{-s = r^ 


(3) 


and 




r3 + s2 = 3r2 


(4) 


From (3), 




$ = r^-r, and s2 = r* - 2 r3 + r2 


(5) 


From (4), 




s2 3z3r2-»^ 


(6) 


Equating (5) and 


(6), and dividing by r2, 








1-2 - 2 r + 1 = 3 - r. 




Collecting, 


etc., 


r2 - r + i r. f , 




whence 




r = 2 or - 1. 




Then from 


(3), 


s = 2. ; 




Therefore 




ic = 4 and 1, 




and 




y = 8. 





247. Squaring (1), we obtain 

^4 _ 3x2^/2 + 2/4 = (3) 

X^-i-2 X2^2 4- ^4 = (^2 + 2/2)2 (4) 

Subtracting (3) from (4), 5 x'^y'^ = (cc2 ^ yiy^ 

Extracting square root, , db xy V5 = x2 -f 2/^ (5) 

Adding (1) and (5), we get 

X2/(l ±V5)=2x2, 
whence x = \ y(l ± V5) (6) 

Putting this value of x in (2), we have 

i 2/2(6 ± 2 V5) + 2/2 = i 2/^(16 ± 8\/5) - ?/. 
Dividing by y^, and multiplying by 4, 

6 ± 2 V54- 4 = J 2/(16 ± 8 V5) - 4 2/, 
or 5±V5 = 2^(liV5) (7) 



214 TABLE BOOK AND TEST PROBLEMS. 

Whence y = ± J V5. 

Dividing (7) by 4, and comparing with (6), we find 

aj = i(5±V5). 



248. (1)- 


xy = 


x-{- 


--hy + -- = 

X " y 


18 


{2)-^xY-- 


- x'- 


^ + 


1 

x^ 




:208 


Adding 4 to both sides, 










h. 


D' 




HJ- 


212 


Put 








hiy 


: ?n, 


and 








Hy 


: n. 


Then from 


(3) and (5) 






m -\- n = 


:18, 


and 








rrfi-^n^^ 


:212. 


From these equations we 1 


find m = 


:14. 


But 








m = 


= X4-1 = 14, 


whence 








X = 


: 7 ± WS. 



(3) 
(4) 

(5) 



Then y is easily found to be 2 ± VS. 

Note. — Neat solutions may be made by putting x + y = s, and xy=p. To make 
these different solutions is excellent exercise. 

249. Dividing (2) by (1), 

x^ + xy(x^ + y^) + xY + 2/* = 121 (3) 

Squaring (1) , x^ + ^^ = 4 + 2 xy (4) 

Squaring (4), x* 4- 2/^ = (4 + 2 x?/)2 - 2 x2?/2, 

or IQ + Wxy -{-2 x-y^ = x* + ?/* (6) 

(3) may now be written 
16 + 16 xy + 2 x22/2 4- Xi/(x2 4- 2/2) + x'^y^ = 121 (6) 

But from (4), x^ -{- y^ = 4: -\- 2 xy. 

Substituting in (6) , we have 

16 + 16 X2/ -f- 2 x^y^ -f X2/(4 + 2 xy) + xV = 121 (7) 

or 5x2y2_^ 20x?/ = 105, 

or x^y^ -f 4 X2/ = 21 ; 

whence xy = 3 or - 7 (8) 



SOLUTIONS TO SPECIAL EXPEDIENTS. 215 

Froin(l), x = 2 + y. 

Substituting in (8), we have y (2 + y) = S ot — 7, 
or y'^ + 2y = S ov -1 ; 

whence i/ = ±2-l=rlor-3, ori v'T^ - 1. 

Hence x = 3, - 1, or 1 ± V- 6. 

250. Dividing (1) by 2, and adding Jx, we get 

y2 __ 4 Vx + Vx V 2/2 — 4Vx + 1 X = X. 
Extracting square root, 

V2/2 - 4Vx + jVx = ± Vx, 
or V ?/2 — 4Vx = jVx or — fVx. 

Squaring, ?/2 — 4Vx == Jx or |x. 

Clearing, etc., 4?/2 = 16 Vx + x or 9x (3) 

Factoring and transposing (2) , we get 

2 V2 ?/ - 2 Vx - 1 = ?/ - Vx + 1 (4) 

(4)x 2= 4V22/-2VX- 1 =2?/-2Vx + 2. 

Subtracting 3, 4V2?/-2Vx- 1 - 3 = 2 ?/ - 2 Vx - 1 (5) 

Put V2^-2Vx- 1 = m. 

Then (5) becomes 4 m — 3 = wi^, 

whence m = 3 or 1. 



Hence V2 ?/ - 2 Vx - 1 = 3 or 1. 

Squaring, transposing, etc. , y = 5 -\- Vx or 1 + Vx (6) 

Squaring (6), ?/2 = (5 + Vx)^ or (1 + Vx)2 (7) 

(7)x4= 4?/2=:4(5 + V^)2or4(l + Vx)2 (8) 

Equating (3) and (8) , 4 (1 + Vx)2 = 16 Vx + x, 
or 4'+ 8Vx 4- 4x = 16 Vx + x. 

Transposing, 3x — SVx = — 4 (9) 

(9) X 12 = 36x - 96Vx = - 48. 

Addmg 64, 36 x - 96 Vx + 64 = 16. 

Extracting square root, 6 Vx = 12 or 4, 

whence x = 4 or f . 

But y = 6 + Vx or 1+ Vx. 

Therefore 2/ = 7, 6|, 3, or If. 

(From Lester B. Fillman.) 



216 TABLE BOOK AND TEST PBOBLEMS, 



261. Let 


a: = to + 2, 


and 


y = z + l. 


Then (1) and (2) become 


x2 + 2/ == 11 


and . 


2/2 + ^ = 7 


Put 


2-i-y = n. 


Then from (4) we have 


x-S = n(2-y), 


or 


X S ^ 

=2 -?/ 

n n 


From (3), 


x2 - 9 = 2 - 2/ 


Equating (5) and (6), 


n n 


or 


n n 



(3) 
(4) 



(5) 
(6) 

(7) 



Adding to each member , we have 

4n2 

71 4^2 n 4^2 

Therefore x- — = ±(s- —\ . 

^ 2n \ 2n) 

Taking the -}- value, we have x = 3. But x = to + 2, hence to = 1. 

RECIPROCAL OR RECURRING EQUATIONS. 

252. Multiplying, we obtain 

a;6 + x5 + X* + 2x3 + x2 + X + 1 = 30 x^. 

Dividing by x^, x^ + x2 + x + 2 + - + i- + ^ = 30. 

X X2 X^ 

Collecting, ^^ + i + x2 + i + x + i + 2 = 30 (1) 

X^ X2 X 

Let y = x + -' 

X 

Then x^ + - = y^-^y, 

x^ 

and x2 + 1 r= 2/2 - 2. 

X2 

Substituting in (1), and collecting, ?/ + 2/^ - 2 2/ = 30 (2) 

(2)x2/= 2/^ + 2/3-2 2/2 = 302/ (3) 

(3)-(2)= y^ - 32/2 = 28?/ - 30. 



SOLUTIONS TO SPECIAL EXPEDIENTS. 217 

Adding 16 y^ + ^f^ to both members, 

Extracting square root, y'^ + V = 4 ?/ + |, 

or 2/'-4|/ = -3, 

whence y = 3 or 1. 

But y = x-{-- = Sorl. 

X 

Therefore x = J (^ ± ^^) OTi(l± V^^) . 

253. Dividing by 8 x^, and arranging, we get 

'^ + ^8-2(«= + i) = ¥ (1) 



Let 








x + - = m. 


Then 








m^ = x3 + - 


and 








x3 + 1 = m3 - 


Substituting 


in 


(1), 


we have 








m^ — 


3 m - 2 m = V", 


or 








m^ — 5 wi = -2/ 


(2) X wi = 








m*- 5m2 = -2/m. 



("i) 



3j». 



(2) 



Adding to both sides -^^ m^ + |-f , we obtain 

m* + f m2 + If = -V-m2 + 2^m + fj^ 
Extracting square root, 



or X 

whence 



254. Dividing by \ll--, Vx+l - 1 = ^^ 7 -^ ^ 
^ ^ Vx 

Squaring, cc + 1 - 2Vx + 1 + 1 = ^-=^. 

X 



m2 


+i 

m 


= 1 


m + h 

TO, 








;2 _ 


la; 

X 


= 2 


1, 

or J. 














(From 


L. 


B. 


FiLLMAN.) 



218 TABLE BOOK AND TEST PROBLEMS, 

Transposing, changing signs, etc., 

2 VSTl = ^ + ^^ "^ -^ = 1 + ic + i. 

X X 

1 2 

Squaring again, 4x + 4 = — h^H \- x- ■\-2x, 

x"^ X 

which may be written 

x2_2 + l-2x + ?+l = 0. 
x2 X 

Factoring, ^x - iV- 2^x - -W 1 = 0. 
Extracting square root, x 1 = 0. 

X 

Clearing and transposing, x^ — x = 1, 

whence x = ^(l ± V5). 

255. Dividing by x + 1, 

X* - x3 + x2 - X + 1 = a(l + x)4 = a(x* + 4x3 + 6x2 + 4x + 1), 
or X* + 1 — ax* — a — x^ — 4 ax^ — x — 4ax + x2 — 6 ax^ = 0. 
Factoring, we obtain 

x^-\-l- a(x* + 1) - x3(l + 4 a) - x(l + 4 a) + x2(l - 6 a) = 0, 
or (x* + 1) (1 - a) - (x3 + x) ri + 4 a; + x2(l - 6 a) = 0. 
Dividing by (1 — a), 

(x4 + 1) _ l±i^(x3 4- x) + ^•^^^ x2 = 0. 
1 — a \ — a 

Dividing by x2, 

x2 \ — a \ X) \ — a 
Put X + i = 2/. 

X 

Then ^2 + 1 = 2/^ - 2. 

X2 



or 



\1 — a/ 1 — a 



SOLUTIONS TO SPECIAL EXPEDIENTS, 219 



,2 ^l + 4a\^, , (l4-4a)2_(l + 4a)2 , l + 4a 



^ ' 1-a j^"^4(l-a)2 4(l-a)2"^ 1-a' 



whence y= 1 + 4a J (l + 4a)-^ +(1 + 4a)(4 - 4a) 
^ 2(1 -a) ^ 4(1 -a)2 



^l 4-4a±\/5(l + 4a) 
2(1 - a) 

Put this = n. 



y = X -\- - = n. 
^ X 



x'^ — nx = — l, 



whence x = i(n ± Vn'^ — 4), 



i^^^ l + 4a±V5(l + 4q) 
4(1 - a) 



o^ (l + 4a)'^ 5(1 + 4a) 2(1 + 4 a) V5(l + 4 a) 
4(1 - ay 4(1 - a)2 "^ 4(1 - a)2 



^/r^-I ^ ^ /lQ(6a - 1) i: 2(1 + 4 a) V5(l + 4 a) 
\ 4ri - a^2 



4(1 - a)' 



Hence K^ ± Vn^ — 4) 



^ l + 4ai:V5(l + 4a)=b2Vl0(6a-l)i:2(l + 4a)V5(l+4a) 

4(l-a) 

256. i^iVse Solution. Squaring and arranging, 

a^x^ + 4 ics 4. 2 a^x* - 4x2 + a2 - 0. 
Dividing by x*, 

a2x* + 4x2 + 2 a2 - — + ^ = 0, 

X2 X* 

a^(x* + l) + 4(x^-i) + 2«2 = (1) 



Put 


-'-h=y' 


Then 


X*- 2 +1 = 2/2. 

X* 


Therefore 


x4 + -\=2/2 + 2. 

X* 



220 TABLE BOOK AND TEST PROBLEMS. 

Hence (1) becomes 
or ahf + 4 ?/ = - 4 a2. 



Therefore y = - — (1 ±Vl - a^); 

that is, x''-l;=:--(l± vT^^) . 

Therefore x^ -h -Al ± Vl - a^)x'^ = 1. 

or x = ±-^-l± Vn^4± V2(l ± VlT^^). 

Second Solution. Squaring, dividing by a^x*, and factoring, we have 

2 



a2\^x2 y Vaj2 

which may be written I x^ -] — | = I x'^ 1 

•^ \ x2y a2\^ x2) 

NOW, (x^ + l)^-(x- 1)^=4. 

Hence, subtracting 4 from each member, we have 

V x2/ a2i^ ^2) 

Completing square, 

(''' " x^ + 0^1''' ~ ^0^ ^» = ^ ~ *• 
Extracting square root, we have 



x^ a^ a2 

or x2-i- = -^(l±Vr=^) 

x2 a2 

Put second member = 2 m2. 

Then x2 - — = 2 m2, 

x2 

and a;*-2m2x2 = l. 



SOLUTIONS TO SPECIAL EXPEDIENTS. 221 

Completing square, 

whence a;^ = rn'^ ± Vl + mS 

and a: = ± Vm^ ± Vl + m^ (1) 

But m2 = -i-(l±Vr:^), 

and m* = ^(1 ± 2Vl _ # + 1 - a*). 

Substituting in (1), we have 

x= ±^| -l(i±vr=^)±^'i + l(i±2VT^^4+i-ao [» 
or x=±W-i±vr=^±V2(i±vr^^. 

2%ir(Z Solution. Dividing by 1 -f x*, 

2x\/l — x^ _a 
1 + X* ~ l' 
By composition and division, 

1 + a;^ + 2 X Vl - a^ _ 1 4- <2 

l + xi-2xVr^^x* !-«' 
which may be written 

x-^(l + a:^) + 2 X Vl - x^ + 1 - a^^ _ 1 4- (1? . 

x2(l4-x2)-2xVl -xi + l-x2 1-a ^ 
Extracting square root, 

X Vl + x^ 4- Vl - x2 _ Vl + g 

X Vl + ^2 — Vl — x2 Vl — a 
By composition and division, 

xVl + x2 _ Vl +'a + Vl - g _ Vl - a^ + 1 ^ 

Vl - x2 vr+a - vn^ « 

Squaring, 

x2 + x* ^ (Vl -^2 4-1)2 ^ 

1 — x^ a^ 

Clearing, 

a^x^ + ^2^4 :::= (Vl - a^ + 1)2 - (Vl - a2 4- 1)2 x2. 
Transposing, etc., 

a2x* 4- 2(Vl - a* 4- 1)^2 = ( VT^^ 4- 1)^. 



222 TABLE BOOK AND TEST PROBLEMS. 

Completing square, 

Extracting square root, 

, vTiT^ + i (vr=^^ + i)(vrTa'^) 

a a 

, (Vl -a--2 + l)VrT«'^ Vl -a^ + l 

or ax^ = ^ ^ , 

a a 

whence x = ± - {(VT'-^^^ + l)(\/l + a^ - 1)}^. 

(From Professor C. Hornung, professor of mathematics, Heidelberg College, Tiffin, O.) 

257. First Solutioii. Expanding, 

a;io _|_ ^9 + x8 + 2 x" + 2 x"^ + 2 x5 + 2 x4 + 2 x^ + x*2 + X + 1 
= 8x^4- 10x^-8x5 + 10x'^ + 8x^. 
Dividing by x^, 

x5 + x* + x^ + 2 X'2 + 2 X + 2 + ? + — + - + - + A: 

Xry'Z ry*0 /y*^ /y»0 

%Aj %Aj %K/ *Aj 

r=8x3 + 10x-^-8 + — + -. 
x^ x^ 

Arranging and factoring, we have 

x5 4--+x*+-+x5 + i + 2x2 4- — + 2X + - + 2 

X^ X* X^ X'^ X 

1\ , ../ o . 1 



X' 



X^ J \ X 



^'+^ -flO X2 + -J-8 



Transposing, 

0,5 4. 1 + 0,4 + 1 . 2 X + ? + 2 33 rf x-^ + ^W 8(x2 + - V 8. 
x^ X* X V xV \ ^ / 

Adding to both sides s/'x + - W s/'x^ + ^\ + 4f x^ + -^ + 4, we liave 

x5 + 1 + 5f x3 + 1^ + lof X + - V ^' + -. + 4^^ + -,1 + ^ 
x^ V ^V V x/ X* V a:-/ 

= 8(. + l)+12(..+ ?i)+12(«' + i)-4 (1) 



SOLUTIONS TO SPECIAL EXPEDIENTS. 223 

Put x-{-- = y. 

X 

Then x^ + — = 2/'^ - 2, and x^ - — = ?/3 - 3 y. 

x^ x^ 

Putting these values in (1), we have 

yo ^ y^ = Sy + 12 (y^ - Sy)-\- 12 {y^ - 2)- 4, 
or r + 2/^ = 8 ^ + 12 ?/3 - 36 2/ + 12 2/2 - 24 - 4 ; 

whence, by factoring, 

t(^y + 1) = 12 y^(ti + 1) - 28(2/ + 1), 
and 2/* = 12 2/2 - 28, or y^ - 12 2/2 = - 28 ; 

whence 2/ = ± Vc ± 2 V2 = x + - (2) 

X 



Clearuig, x^ - xV6 ± 2\/2 = - 1, 

whence x = ±K^^±2V2 ±V2 ±2\/2). 

Note. — A better solution is given below. 

Second Solution. Expanding and refactoring, without transposing 
terms, 

(xio + 1) + (x9 + x) + (x8 + x2) + 2(x" + x^) + 2(x6 -f x4) 4- 2 x5 

= 8(x8 + x2) 4- 10(x' -f x3) - 8 x5 (1) 

(1) -f- x^ = 

= ^^*' + ^b) + 10(^^ + ~)-8 (2) 



Put 


x + i = 2,. 


Then 


x^ + l = 2/^-2, 




x8 + l = 2/3-3 2/, 



x4 + ^ = 2/*-42/2 + 2, 

X* 

and x^ H — — ^5 _ 5 ^/S _^ 5 ^, 

x^ 

Substituting these several values in (2), we obtain by transposition, etc., 
2/5 4. 2/i _ 12 2/3 - 12 2/2 + 28 2/ + 28 = (3) 



224 



TABLE BOOK AND TEST PBOBLEMS. 



Since by changing the signs of the second and every alternate term 
in (3) the algebraic sum of the coefficients would be zero, by the theory 
of equations — 1 must be a root of equation (3). Therefore it is divisi- 
ble by 2/ + 1. 

Performing the division, 

2/* -12 2/2 + 28 = 0, 



2/=±V6 ±2\/2. 
1 

X 



whence 
Therefore x H- ^ = — 1 

-h- = ±V6d=2V2 



and 



X 



(4) 
(5) 



From (4) we find 

iVzra-i 

X = — 

2 
From (5) we find 

x = ± jVe ± 2 V2 + i V2 ± 2\/2. 

Therefore x has 6 imaginary values, and 4 real ones. The imaginary 
values are, — 



V-3-1 



_V-3-l 



X = ' 



jV(6 + 2V2)+ jV (2 - 2\/2), 
jV(6 + 2 V2)4- i V(2 - 2 V2), 
iV(6 - 2\/2)+ jV(2 - 2V2). 



_ 1 V(6 - 2V2) + ^V (2 - 2 V2) 
The real values are, — 



X = 



jVg + 2 V2 + J V2 4- 2V2 = 2.5843176. 



jVe - 2 V2 + jV2 + 2 V2 = 1.9891296. 
iV6-2\/2 + jV2 + 2\/2 = .2082386. 



- h'-Ve + 2\/2 + iV2 + 2V2 



.3869493. 



(From Professor B. F. Burleson, Oneida Castle, N.Y.) 



SOLUTIONS TO SPECIAL EXPEDIENTS. 225 

HIGHER EQUATIONS. 

258. Squaring the given equation, we obtain 

2V[(^^ - «'^)(^^^ - &')]- 2V[(^^ - c2)(x2 - cZ^)]=z: a2+ 62 _ c2 _ d^ (1) 
Squaring (1), etc., we have 

8-v/[(^^^ - «'^)(^^ - ^■^)(^'^ - c^)(^^ - ^^)]= 8^4 - 4(a2 + 62 + c2 + cZ2)x2 
+ 4 a262 + 4 c2cZ2 - (a^ + 62 - c2 - ^2)2 (2) 

Squaring (2), canceling equal terms in opposite members, refactoring, 
etc., we obtain a pure quadratic from which we find that 

X =± V{(«H^^- c2-cZ2)l-8(a262 + c2dJ2)(^2_|_52 _ ^2 _ cPy2 

+ 16(a262 _ c^d^y2} -^ 2v'{B(a-62 - c2cZ^)(a2 + 62 - c2 - (^2) 

_ 2(a2 + 62 + C2 + fZ2) (^2 4. ^2 _ c2 _ (^2)2j 

= (when a = 36, 6 = 40, c = 13, and d = 61)± 85. 

The formula found for the value of x is evidently also that which will 

be found for any equation obtained by per mutating the signs of the given 

equation. To illustrate : if a = 375, 6 = 408, c = 180, and d = 297, we 

find by the formula that x = ± 425, which are the roots only of the 

equation . . 

V x2 — a2 _ Vx2 — 62 = V x2 — c2 — vx2 — d^. 

Again, if a = 94, 6 = 95, c = 96, and d = 20, we find by formula that 

/y.__Ll /r 2 072 25262'706962 5^ 
^— ±2Vl 5124TT74646 S 

= ± 100.549351213668953, 
which are the roots only of the equation 

Vx2 - a2 + Vx2 - 62 + Vx2 - c2 = Vx2 - d^ 
(correct to 15 decimal places). (From B. F. Burleson.) 

259. Clearing from the radical, we obtain, by factoring, 

16m4(l - x2)2 _ 8x2(2 - m*) (1 - x2) = - m^x^ (1) 

Considering (1) as a quadratic equation whose unknown quantity is 
(1 — x2), we find by its resolution that 



V 4m* J 

ELLWOOD^S TEST PROB. — 15. 



226 TABLE BOOK AND TEST PBOBLEMS. 

which is also a quadratic equation whose unknown quantity is x^. Hence 



^2 ^ ±2m'^\-\^2± 2(1 - m^)l - m^ ^ 
2 ±2Vl -m^ -m^ 

Reducing this value of x^ to its lowest terms, we have 

^2 = 2m^ 



whence x = 



m2±V(2±2Vl-m4) 

± m\/2 

±A/lm2 ± V[2 ± 2 V(l - m*)] ] 

(From Professor B. F. Burleson.) 

Note. — In 1879, F. P. Matz, A.M., Ph.D., published this as a prize problem in 
Barnes's " Educational Monthly," and awarded the prize to Mr. Burleson of Oneida 
Castle, N.Y., for the solution given above. 

260. Raising to the fourth power, we have 

16x^-16x6 =(2 -0^2)4. 

Now, (2 + x2)4 = (2 - x^y -f 16 x6 -f 64 x\ 

Hence, adding to each member 16 x^ + 64 x^, we have 

16x^ + 64x2 =(2 + x2)4 =(4 + 4x2 + x^y. 

Put 4 + 4 x2 + X* = ?/. 

Then the equation becomes 

or 2/2 - 16 2/ = - 64, 

and 2/ = 8- 

Then x* + 4x2 -f- 4 = 8, 

x2 + 2 = V8 , 



x=V2(V2-l). 

261. Put Vx = y^. 

Then the equation becomes y^ — y'^ = 100 (1) 

(l)X2/= y^-y^ = 100y (2) 

(2) + (1) = 2/* - 2/2 = 100 2/ + 100. 



SOLUTIONS TO SPECIAL EXPEDIENTS. 227 

Adding to each member 16?/2 + ^|-^, 

2/* + 1^2/^ + -i- = 1^2/^ + 100?/ + ^1^. 

2/2-42/4-4 = 5 + 4, 
whence 2/ = 5, 

2/3 = 125 = vx. 
Then x = 1252 = 15,625. 

262. ractoring,8^x-^Vx2 + 1 + 1^ = 0. 

Therefore x — J = 0, and x = J, the real root. 

x2 + ^ + ? = 0, or x24-^ = -5. 
2 4 2 4 

Completing square, ^2 + - + — = — -- = -—. 

^ ^ ^ ' 2 16 16 4 16 



Extracting square root, x + J = ± i V— 35, 



and X = — J lb JV— 35 = J(— 1 ± V— 35), the imaginary roots. 

263. x3 + 2x2 + x = 18 (1) 
(1) X X = X* + 2 x3 + x2 = 18 X (2) 

(2)- 2 x(l)= x*-3x2 = 20x-36. 

Adding 16 x2 + ip, 

X* + 13x2 + i|^ = 16x2 + 20x + -2;^. 

Extracting square root, x2 + -^^ = 4 x + |, 

whence X = 2. (From Dr. I. J. Wireback.) 

264. Let _L_=:i7^2, 

1 + x 



Then the equation becomes 



A/m2Vm2^^^v^. 
12 



m2 2x 1 



144 1 + X 
Clearing, etc., x2 + x = 72, 

whence x = 8 or — 9. 



228 TABLE BOOK AND TEST PBOBLEMS. 

265. This may be written 

x3 + 3x2-60^-8 = (1) 

Put X = 2/ — 1. 

Substituting in (1), we have 

2/3 _ 9 2/ = 0, or ?/(?/2 _ 9) = 0. 
Hence ?/ = 0, and y^ - 9 = 0, y^ = 9, y = ± S. 
But x= y — 1: consequently x = — 1, 2, and — 4. 

266. Transposing, x^ - 6 x^ + 11 x - 6 = 0. 
Factoring, (x - 2) (x2 - 4 x + 3) = 0. 
Therefore x — 2 = 0, 

or X = 2, 

and x2 - 4 X 4- 3 = 0. 

Adding 1 to both members, x2 — 4x + 4 = l. 

x-2 = ±l. 
x = 2±l = 3orl. 
Hence x has three real roots : 1,2, and 3. 

267. Adding 4 to each member, and factoring, 

(x2-3x-2)2=64. 
x2-3x = 2±8 = 10 or - 6. 
Completing square, x2 — 3 x + | = -^^ or — ^^-. 

X - f = ± I and ± jV- 15. 

X = 5, -2, and ^(3 ± V- 15). 

268. Multiplying by x, x* + 16 x2 = 128 x. 
Adding 16 x2 + 256, we have 

X* + 32x2 + 256 = 16x2 + 128x + 256. 

x2 + 16 = 4x+ 16. 

x2 = 4 X. 

x = 4. 

(From Dr. I. J. Wire back.) 

269. Put \/x = ?/. 
Then y^ - y = 14. 
Adding 2 to each side, and factoring, 

(2/2 + 4)(2/ + 2)(2/-2)=2/-2, 
or (^;2 + 4)(^ + 2)=l (1) 



SOLUTIONS TO SPECIAL EXPEDIENTS. 229 

But (2/2 4. 4) (2/ + 2) >1. 

Therefore (1) is an absurdity, which could only be brought about by 
using a zero factor. 

Hence y — 2 = 0, 2/ = 2, and x = 4. 

(From Dr. I. J. Wireback.) 

270. Adding x* + 2 x^ to each side, we have 

x* + 2 x3 + x2 = x^ + 2 x2 + 1. 

X2 + X = X'2 + 1, 

whence x = 1. 

271. Multiplying by x, x* — 6 x^ + 4 x = 0. 

Adding 4 x^ + 1 to each member, and transposing, we have 

x*-2x2 + l=4x2-4x + l; 

whence x^ — 1 = 2 x — 1, 

x2 = 2 X, X = 2. 

(From Lester B. Fillman.) 

272. 0:3-3x2 + 4 = (1) 
(l)x x = x*-3x3 + 4x=:0 (22) 
(2) - 3 X (1) = X* - 9 x2 = - 4 X - 12. 

Adding 16x^ + -^^, x* + 7 x2 + -\^- =: 16x2 - 4x + J. 
Extracting square root, x^ + J = 4 x — J. 

Transposing, x^ — 4 x + 4 = 0, 

whence x = 2. 



273. 


V^-l^x = 100 


0) 




x6 -x^ = 100 


(2) 


Let 


1 
x^ = y. 




Then (2) becomes 


yZ _ if = 100 


(3) 


(3)X2/ = 


2/4-2/3 = 100 2/ 


(4) 


(4) + (3) = 


2/4 _ 2/2 zr: 100 y + 100. 




Adding 16 2/2 + 225^ 


2/4 + 152/2 + 22A = l6y2^ 100 2/ + ^F. 




Extracting square root. 


2/-^ + -¥- = ±(42/ + -V-). 




Taking + value, 


2/2 -42/ + 4=1 9, 

2/ = 5 or — 1. 




Therefore 


X* = 5 or — 1. 
X = 56 and - 16 r= 15,625 and 1. 





230 TABLE BOOK AND TEST PBOBLEMS, 

274. This equation may be written thus : — 

9x^ - X* - 18x3 - 2x'^ + 45x-2 - 25x2 - 36x - 24x = 144 - 36. 
Transposing, 

9x* - 18x3 + 45x2 _ 36x + 36 = X* + 2x3 4- 25x2 + 24x + 144. 
Extracting square root, 

3x2-3x + 6=rx2 + x + 12. 
Transposing, 2 x2 — 4 x = 6, or x2 — 2 x = 3. 

Adding 1, x2 — 2 x + 1 = 4, 

whence x = 3 or — 1. 

(From Dr. I. J. Wireback.) 

275. x3 + x2 = 80 (1) 
Multiplying by x, x^ + x^ = 80 x (2) 
(2)-(l)= x*-x2 = 80x-80. 

Transposing, x* + 80 = x2 + 80 x. 

Adding to each member 24 x2 + 64, we have 

X* + 24 x2 + 144 = 25 x2 + 80 X + 64 ; 
whence x2 + 12 = 5 x 4- 8, 

or x2 — 5 X =: — 4, 

and X == 4 or 1. 



276. Transposing and squaring, 



ifx-2a-^] = (l 

X2V X 



--V 



Arranging, A £Z _ !| = ( i _ « V (1) 

X x^ V X^J 

Let y = ^--2 

X2 

Then (1) becomes 2/^ - ^^ = _ ?-^, or x'^if -xij =-2a. 

Putting xy = m, we have 



X X' 



m2 — m — — 2 a, 



whence m = \(\ ± Vl — 8a). 
But 2/ = 1 ;• .'. xy = x — = m, 

X2 X 

m , 1 



whence x = — ± 7 V4 a + m^. 



SOLUTIONS TO SPECIAL EXPEDIENTS. 231 

Substituting value of ??^, we have 

X = i j ] ± VI -8a ±V2±2(l-8a)^+8a | . 

277. Squaring, ^ + ^""^ ~ ^ rz: x'^ - 4a: + 4. 
X — Vx"^ — 9 

By composition and division, 

2x x2-4x + 5 



Squaring, 
By division, 



2Vx2"^^ ^'^-4x4-3 

x2 ^ x^ - 8x3 + 26x2 - 40x + 25 

x2 - 9 x^ - 8 x3 + 22 x2 - 24 X 4- 9 * 
x2 x* - 8 x3 + 26 x2 - 40 X + 25 



Extracting square root. 
Clearing and collecting. 



9 4(x2-4x4-4) 

X _ x2 — 4 X + 5 ^ 
3~ 2(x-2) 



x2-8x = -15, 
whence x = 3 or 5. (From Lester B. Fillman.) 

278. Reducing fractions, 

^ + ^ + ^-"^ =a. 

1 - X + X2 1 + X + X2 

Clearing, etc. , 2 + 4 x2 = a + ax + ax*, 

or ax^ + (a — 4)x2 = 2 — a. 

Completing square, etc. , we find that 



whence 



2a ~^^ 4a2 ' 



X =a/^(4 -a± jVlO - 3 a2). 

(From Lester B. Fillman.) 

279. It is easily seen that the first member is equal to (x + Vx)'^. 
Hence the equation may be written 

(x + Vx)2 = 32(x + Vx) - 240. 
Completing square, 

(x + Vx)2 - 32(x + Vx) + 256 = 16. 

Extracting square root, x -f Vx — 16 = ± 4, 
or X 4- Vx = 20 or 12. 

Hence Vx = 4, or — 5, or 3 ; 

and X = 16, 25, and 9. 

Note. — To satisfy the equation with x = 25, we must use >/x = — 5 instead of v^ = 5. 



232 



TABLE BOOK AND TEST PROBLEMS. 



280. First Solution. Clearing, 

x2-2xVx- 7a: = 8V^- 16. 
Adding 8 x^ x^ — 2 x Vx + x = 8 x + 8 Vx — 16, 

which may be written x^ — 2 x Vx + x = 8 x — 8 Vx — 16 + 16 Vx. 
Factoring, then transposing, 

(x - Vx)2 ^ 8(x - Vx) + 16 = 16 Vx. 
Extracting square root, x — Vx — 4 = 4 Vx, 
or X = Vx + 4 Vx 4- 4, 

and Vx = Vx + 2, 

or 





Vx- 


-Vx + i = f. 

Vx = 2 and — 1, 


Therefore 




X = 16 and 1. 


Second Solution. 


Let 


Vx = 2/. 


Then 


] 


/ ^=«. 



2/ - 2 y^ 
Clearing and transposing, 

^4 _ 2 2/3 - 7 ^2 _ 8?/ + 16 = 0. 

Adding 16 y to both members, 

2/4 _ 22/3 - 7?/2 + 8?/ 4- 16 = 16?/. 

2/2 - 2/ - 4 = 4 V^. 

2/2 = 2/ -K 4V^-h4. 
2/ = ^2/ + 2. 
V2/ = m. 



Let 
Then 



Therefore 
and 



m 



m2 - m = 2. 
2 _ m + 1 = f . 

77^ = 2 and — 1. 
2/ = 4 or 1, 
X = 16 or 1. 

(From Dr. I. J. Wireback.) 



Third Solution. Clearing, and freeing from radicals, we obtain 

X* - 18x3 + 49x2 - 288x + 256 rr (2) 

Factoring (2), 

(x2 - 17x + 16)(x2 - X + 16)= (3) 



SOLUTIONS TO SPECIAL EXPEDIENTS, 233 

Therefore x'2 _ 17 ;;c _l_ 10 = (4) 

and x2 - X + 16 = (5) 

From (4) we find x = 1 or 16, 

and from (5) we find that x = ^1 ± SV^^). 

(From Professor B. F. Burleson.) 

Fourth Solution. Put Vx = y. 

..7 8 



Then 
Dividing by 8, 



or 



y -2 if 

y 7 ^1 

8 8(2/ - 2) y^' 

y^-2y-7 ^ 1 
8(2/ -2) "2/^' 



Adding to both members, 

y -2 

y^~2y + l ^1 _1_ 
8(2/ -2) ?/2 2/-2 
Clearing, etc., 2/2(^/2 _ 2 ?/ + 1) = 8(?/2 + 2/ - 2). 

Factoring, y^(y - 1) (?/ - 1) .= S(y _ 1) (2/ + 2). 

?/3_t/2^82/ + 16 (1) 

Multiplying by y, y^ = tf -^ Sy'^ -^ 16y (2) 

Adding (1) and (2), ' ?/* = 9 ?/2 + 24 ?/ + 16. 

y^ = T(Sy + 4); 

whence 2/ = 4, - 1, and - I ± i V^^. 

Therefore y^ or x = 16, 1, and ^1 ± 3V^^). 

Fifth Solution. Let y = Vx. 

Then "^' y- '^ zz J-. 

2/ - 2 ^2 
Clearing and transposing, 

Factoring, 

(2/-4)(2/-l)0/ + 32/ + 4)z.O. 
Hence ?/ - 4 = 0, ?/ = 4, and x = 16. 

?/ - 1 = 0, ?/ = 1, and X = 1. 
2/2 + 3?/ + 4 = 0, ?/ = - J(3±V^), andx = Kl±3A/^). 



234 TABLE BOOK AND TEST PROBLEMS. 



281. This may be written 



jax'^ — a , 



x^-a 
x"- ^ x^ 



- Vax'-^ — a -j- - Vx-^ — a = x^. 



or 

X X 



Multiplying by x, 



Vax''^ — a + Vx* — a = x^. 
Transposing and squaring, ax^ — a — x"^ — 2 x^ Vx* — a + ic* — a, 
whence, dividing by x^, we have a = x* — 2 xVx"* — a -\- x'^. 
Transposing and squaring, 

4x2(x*- a)=x8 + 2x6 4-(l-2a)x*-2ax2+ a^. 
Collecting terms, and dividing by x*, we get 

x*-2x2 + l-2a + ?-^ + ^ = 0. 

X^ X* 

Combining, 

Extracting square root, x^ — -- — 1 = 0; 



whence x* - x^ = a, and x = ± V ^ ± \ V4a + 1. 

282. (1) - x^?/ = 5c^ 4- i = 2/ + - (3) 

x^ 2/ 

(2)- 3x?/3 = 1 f2/3 + i) =: Sf X + 1) (4) 

Put 

Then 

Let 

Then ,/ + 1 = ^3 _ 3 ,^. 

Adding (3) and (4), and substituting, we get 

723 — 3 jjj3^ or 71 = m V3 r= [ X H- - J V3. 
Substituting in (3), we have 

X^ \ XI 



x^ + i = 
a;' 


-J 


-?)= 


K-a 


X 


: m. 


*+5 = 


m^ -3 m. 


" + 5 = 


: 72. 



SOLUTIONS TO SPECIAL EXPEDIENTS. 235 

Dividing by /x + i V x'^ - 1 -{- - = \/S (6) 

\ xj x^ 

Completing square, x^ — 2 + -^ = v3— 1, 

whence x-i = (v^-l)* (6) 

Completing square again by adding 3 to (5) , 

x2 + 2+l = ^3 + 3, 

X"' 

whence x + ^ = ( \/3 + 3) (7) 

From (G) and (7), we easily find x = J{(3 + \/3)2 4-(v^ _ 1)^}. 
Then y = i{\/3(v^3 + 3)*±(3v'9 - 1)*}. 

283. Let 2 X + 3 =: m^, and 2 x - 3 = n^ 



Then ??l+«=4/«|^«3x . ^^^ 

Clearing, etc., 13 wP-Tf-^m"^ + /i^) = 4 m^ + 4 n^ (2) 

or 13 m^n^ = 4 7n* — 4 m2722 + 4 iz* (3) 

Taking ^m'^n'^ from both members of (3), 

9 mhi'^ = 4 m* — 8 ^h^w^ + 4 tz*, 
whence 3 ??in = 2 771^ — 2 ?i2 rr 2(7>i2 _ ^^2^ ^4^ 

Adding 12 mV to both sides of (3), 

25 m^n'^ = 4 m* + 8 mhi^ + 4 7i*, 
whence . 6 mn = 2 (m^ + ?z2) (5) 

From (4) and (5), m = 2 ?2, and m^ = Sn^ (6) 

Substituting values of m and n in (6), we get 

' 2x + 3r:.8(2x-3), 
whence x = f I 

^6' 



284. Factoring, 

Dividing by x*, 

V Sx) x^y 3xJ X* 



236 TABLE BOOK AND TEST PROBLEMS. 

Putting (1 H 1 = 2/5 the equation becomes 

V SxJ 

3 70 

x^ x^ 
Completing square, ^/^ - ^ 2/ + A = ^ + A = 7^' 

• / 



Extracting 


square 


root, 






y 


3 

2x'^ 


-s-=s»-^.- 


But 






3a; 


Therefore 









whence x = J (— 1 i v — 251) and 3. 

285. Developing, i -^ x'^ = a -h Sax + ^ax^ -^ ax^. 

Dividing by x^, ]^ ^ x = -^^ -}-^- + S a + ax. 

XP' X^ X 

Factoring, 1.j^x- a[ — ^x\^ 3af-+ 1 

X?' Vx^ J \x 

Dividing by - + 1, we have 

X 

x + i- 1 rra(x + i^l)+3a (1) 

X X 

(1) may be written x ^ l = a(x + - — 2) + 4a. 

X V ^ / 

Subtracting 1 from both sides, and factoring, 

/ Vx - -^y = a( Vx - — )% 4 a - 1, 

V Vx/ V Vx/ 

Put /vx-— y=?/2. 



Then (2) becomes ay'^ — i/'^ = 1 — 4: a. 

1 -4, 
a- 1 



1 — 4 a 
?/2 = = 9?i'2, suppose. 



Therefore ( Vx ) = m"^, and Vx = 

\ Vx/ Vx 



m. 



SOLUTIONS TO SPECIAL EXPEDIENTS. 237 

Clearing and transposing, x — mVx = 1. 
Completing square, 

4 4 4 

Extracting square root, Vx = db - V4 + ??^^ ; 

whence Vx = ^ (m ± V4 + ni^) , 



and X = J(m2 ± 2 ?n\/4 + m"^ 4- 4 + ?7i2), 



or X = J(m2 ± mV4i -{- m^ + 2). 

Substituting value of m, and reducing, 

2(1 - a) 

286. Multiplying by x, 

x*-6x3 + llx2-6x = (1) 

The square root of the left member is x^ — 3 x, with a remainder of 
2 x^ — 6x. Hence (1) may be written 

(x2-3x)2-f2(x2-3x)=:0. 

Completing square, 

(x2 _ 3 x)2 4- 2(x2 - 3 x) + 1 = 1. 

Extracting square root, x2 — 3x + l = ±l; 
whence x2 — 3 x = or — 2, 

and X = 3, taking value. 

Taking the other value, x = | ± ^ = 2 or 1. 

287. First Solution. 

Clearing, x2 - 5 x = 12 + 8 Vx. 

Adding x + 4 to both sides, 

x2 - 4 X + 4 = 16 + 8\/x 4- X. 
X - 2 = 4 -f Vx. 
X — Vx = 6. 

X - Vx 4- i = y. 

vs - } = d= |. 

Vx = 3 or - 2. 
Therefore x = 9 or 4. 



238 TABLE BOOK AND TEST PBOBLEMS. 

Second Solution, Put Vx = y. 

Then, clearing, etc., 

2/4 _ 5 2/2 _ 8 ?/ _ 12 = 0. 
Factoring, 

(2/ + 2)(?/-3)(t/2 + 2/ + 2)=0. 

Then 2/ + 2 = 0, y = -2. 

2/ - 3 = 0, y = Z, 

2/2 + ?/ + 2 = 0, y=:^(l±V^l). 

Substituting value of Vx, we find that 

X = 9, 4, and J(- ^ ± V^^). 

288. i^iVs^ Solution. 

Let x = m + w, and y =z m — n. 

Substituting in (1) and (2), the equations become 
2 m^ + 6 mn^ = 72, and 2m^-2 imi^ = 48. 
From these we find m = 3, and n = l. 
Therefore x = 4, and y = 2, 

Second Solution, To (1) add 3 x (2). The sum is 

x3 + 3 x^y + 3 x?/2 + 2/3 = 216 (3) 

Extracting cube root, 

X + 2/ = 6, or X = 6 — 2/ (4) 

(1)^(4)= x2-x2/ + 2/2 = 12 (5) 

(4)2 _ (5) = 3 X2/ = 24, or xy = 8. 

Substituting value of x from (4), we get 

(6-2/)2/ = 8; 
whence 2/ = 2 or 4, 

and X = 4 or 2. 



289. First Solution, Let x -\- y = s, and x?/ = p. 
From (2), s = Vl3 + 2p, 

From (1), s = 



Equating, \/l3 -\- 2p = 



13 -p 
35 



13 -i? 

whence 2p^ - 39p2 + 972 = 0, 

or 2p^ - 36 p2 _ 3^)2 4. 972 = 0. 



SOLUTIONS TO SPECIAL EXPEDIENTS, 239 

Factoring, 

2i>2 {p _ 18) - 3(j9 - 18) (2) + 18) = 0. 
Reducing, 2p'^ = ^{p^ 18), 

whence j) = 6 or — |. 

Adding 12 (or 2^) to both members of (2), 

-ic2 + 2^; + ?/2 = 25, 
whence a^ + ?/ = ± 5. 

Subtracting the same from both members, we find 

x-y=l. 

Therefore ic = 3, 

and 2/ = 2. 

Second Solution. 

Factoring (1), (x + y) {x^ — xy -\- y'^) = 35, 

or x'-xy + y'^ = -^^ (3) 

x-\-y 

(3)-(2)= _a:y = _13+ ^^ 



Multiplying by — 2, 2 iC2/ = 26 — 

Adding (2), x^ + 2xy + y'^ = S9- 

or (a: + yy = 39 - 



70 



a; 4- 2/ 

70 
x + y 

70 



x + 2/ 
Multiplying hj x -{- y, (x + 2/)^ = 39(x + i/) — 70. 

Put , X + y = m. 

Then 7l^3 _ 39 ^ ^ _ 70. 

Multiplying by m, m^ — 39 m^ = - 70 m, 

which may be written m^ — 14 m^ = 25 m^ — 70 m. 

Adding 49 to both members, 

m* - 14 m2 + 49 = 25 7^2 - 70 m + 49. 

7?l2 _ 7 — 5 ,7^ _ 7. 

m = 5 = a: + 2/ (4) 



240 TABLE BOOK AND TEST PBOBLEMS, 

(4)2 = x2 + 2 x?/ + 1/2 = 25 (A) 

(A) - (2) = 2xy = 12, xy = 6 (B) 

(^) - 4 X (B) = 0^2 - 2 a:?/ + ?/2 = 1. 

x-y =1. 
Therefore x = 3 or 2, i/ = 2 or 3. 

(From Lester B. Fillman.) 

Third Solution, x'^ + y^ = 13 (1) 

x3 -f 2/3 = 35 (2) 
Let X = m + j9, y = m — p. 

Then 2 m2 + 2^)2 :^ 13 (3) 

and 2 7?i2 4. 6^2 ^ _^ (4) 

/^\ /ON ^9 35 — 13 m ,_. 

(4) -(3)= 4i)2:^ _ , (5) 

Substituting value of 2^2 [j^ ^3^^ ^e have 

o o , 35 — 13 7?i -,0 
2 m2 -] =r 16. 

2 77i 

Clearing, 4 m^ + 35 — 13 m = 26 m, 

or 4 m^ — 39 m = — 35. 

Multiplying by m, 4 m* — 39 7?^2 = — 35 m. 

Adding to each side 4m2 + (-\^-)2, we have 

4 m* - 35 m2 + (-^f-y^ 4t m^ - 35 ?>i + (¥)^- 
Extracting square root, we have 

2 m2 - -y = ± (2 m - ^^-), 
whence m = 1 or 2J. 

Substituting m = 2} in (5), we find 

But X - m +j9 = 2j4-J = 3, 

and ?/ =z m — j) = 2^ — J = 2. 

(From Dr. I. J. Wireback.) 

Fourth Solution, 

Let s = x -\- y^ X) = xy. 

Then (1) and (2) become s^ - 3 sp = 35 (3) 

§2 _ 2p = 13 (4) 

Eliminating sp, we obtain s'^ — 39 s = — 70. 



SOLUTIONS TO SPECIAL EXPEDIENTS. 241 

Multiplying by s, s* — 39 s^ r= — 70 s. 

Adding to both sides 25 s'^ + 49, we get 

s4 _ 14 52 _^ 49 ^ 25 s'^ - 70 s + 49. 
s2-7=±(5s-7). 
Taking the + value, we have 

s'^ r= 5 s, or s = 5 = X + ?/, 
whence x and y are readily found. (From Educational Neius.) 

290. First Solution. Developing and subtracting (1) from (2), 

2xy(x + y)= 420 (3) 

Dividing 2 x (3) by (1), and reducing, 

4txy _84 
(x — y)'^ 16 

By composition, iHL±jO! ^. WO. 

(x — ijy^ 16 

Evolving, 21 + 1^10. 

X — ?/ 4 
This may be written x + ?/ = 10, 

X - 1/ = 4 ; 
whence X = 7, and ?/ = 3. (From Dr. I. J. Wireback.) 

Second Solution. Developing (1) and (2), we obtain 

x3 - x^y - xif + y^ = 160 (3) 

x^ + x'^y + xy'^ -\-if = 580 (4) 

Adding (3) and (4) , and dividing by 2, 

x3 -f ?/3 zr 370 (5) 

Subtracting (3) from (4), we get 

2 x2?/ + 2 xif = 420 (6) 

Multiplying (6) by |, 3 x'^y + 3 x?/^ =r 630 (7) 

Adding (5) and (7), 

x3 + 3 x2?/ + 3 xy^ + ?/3 ^ 1000 (8) 

Extracting cube root, x + ?/ = 10 (9) 

Factoring (6) , and substituting value of x + ?/, 

2 X2/ = 42 (10) 

Subtracting 2 x (10) from (9)2 gives 

x2 - 2 xy + ?/ = 16, X - ?/ = 4 (11) 

From (9) and (11), x = 7, and y = S. 

ellwood's test prob. — 16. 



242 TABLE BOOK AND TEST PROBLEMS. 

Third Solution. Subtracting (1) from (2) gives 

2xy(x^ 2/)=420, 
420 









2xy 


x + y 


From (2), 






x2 -f y^' : 


580 
X4- ?/ 


Adding (3) 


and (4), 


x2 


+ 2 X?/ + ^/'^ : 


1000 

X + ?/ 


Clearing, 






(X + yy : 


= 1000. 


Extracting ( 


3ube root, 




X + ?/ : 


= 10 


Substituting in (2), 




x-2 + y^ : 


= 58 


Subtracting 


(6) from 


(py 


2, xy 


= 21 


From (5), 






X 


^ 10 - y. 


Substituting in (7), 




2/(10-?/) 


= 21, or ?/2 


Completing square, 


y 


- 10?/ + 25 


= 4, 


hence 






y 


= 7 or 3. 


Hence 






X 


=- 3 or 7. 



(3) 
(4) 



(5) 
(7) 



10?/ --21. 



291. Let X + 2/ + ^ = s, x?/ + x^ + 2/.^ = m, and xyz = n ; also, for con- 
venience, put a-\-'b-\-c = d = 216, ab -\- ac -\- he = g = 13248, and 
a&c=i) = 207360. 

(1) -I- (2) + (3) = [by involving terms and factoring] ns = d (4) 

Similarly, (1) x (2) + (1) X (3) + (2) x (3) = 4 m?i2 - n'^s'^ - g (5) 

and (1) X (2) x (3) = 4 smn^ - nH^ -^n^-p (6) 
(5) X 715 — (6) = 8 ?i* = nsg — p =^ dg — p. 

Therefore n = i '^[^(dg - p)^ (7) 
Now, the given equations are equivalent to 

ns — 2nz = a (8) 

ns-2 ny = b (9) 

7is — 2 nx = c (10) 

Substituting in (8), (9), and (10) the value of 7^5 as given in (4), and 
the value of n as found in (7), and resolving severally as simple equations, 
we find that 

d — a &4-C _ A 



</['Kdg-p)2 



b ■}- c 




■y/[2(a -{- b -\- c)(ab + ac + bc)- 


- 2 abc] 


a -\- c 




^[2(a + & + c) («6 4- ac + be) - 


- 2 abc'] 


a-\-b 





y _ d — b u T '^ 3 

d — 6 te- -r (7 o 



^[2(dg-p)^ ^[2(a -f ^ + c)(ab + ac + ftc)- 2a?;c] 

(From B. F. Burleson, in yotes and Queries.) 



SOLUTIONS TO SPECIAL EXPEDIENTS. 243 

292. Simplifying, we liave 

- Vx^ — b'^ + - Vx- — a'^ = Vx — d^ -\- ~ Vx'^ — c'^. 

JO tJU *Ay «/y 

Clearing, aVx^ — b'^ + bVx'^ — a^ = cVx-^ — d^ + cWx- — c^. 
Transposing and squaring, 



a^x^ - a'b^ - 2 acV\¥ - {b^ + d^)x^ + b'^d^} + c'^x^ - cH^ 



= d-x^ - d^d^ - 2 MV{xi-(a-2 + c^)x--^ + a'-c'] + ^^x'^ - a-¥-. 
Collecting and canceling terms, 

{aP' + c-2)x2 - 2 ac^/{^' - (b'^ -f c^^x^ + b'^d^} 

= (d^ + 6-^)x2 - 2 6d V{xi - (a^ + c^x^ + a-^^}. 
Transposing, 



= - (a2 + c2)x2 - 2 6d V{x* - (a2 + c 0x2 + ^-Jc-ij. 

We can form two perfect squares by adding to each side of the equation 
the fourth power of x and the square of the half of the coefficient of each 
radical ; that is, x^ + (ac)2 + (bd)'^. Performing this addition, we have 



{X* - (62 + cZ2)x2 + 62cZ2} - 2 acV{x^- (62+ C?2)^2 j_ ?>-2^2}-f a2c2 

=|xi - (a2 + c2)x2 + a2c2} - 2 bdV{x^ - (a2 + c2)x2 + a2c2}+ b'^-d^ 
Extracting square root, 



V{x±-(62 + d^)x2 + 6-'cZ2} ~ ac = ± [\/{x^ - (a2 + c2)x2 + a2c2}-6c/]. 
Using the minus sign, and transposing ac, we obtain 



V{xi-(62 + d2);^2+52^2| 3^ _ V{xi - (a2 + c2)x2 + a2c-2} 4- (ac + 6(^). 
Squaring, canceling, and collecting, 
(a2 - 62 + c2 - c?2),^2 _ 2 ac{ac + 6(7) 

= - 2(ac + 6cZ) V{x^ - (a2 + c2)x2 + a^c:^}. 
Squaring again, dividing by x2, canceling and collecting, 
4(ac 4- 6(i)2x2 - {aP- + (fi - 62 - cZ2)2^2 

= 4(a2 + c2) (ac + 6d)2 - 4 ac<iac + 6(7) (a2 + d^ - 62 - (^2) ; 
whence 

^ -^ / 4(a2 -I- c2) (-(^c ^ ^^;)2 _ J^:(ac + 6cZ) (a^ + cP - b' - d^) 
^ 4(ac + 6c02 - (a2 + c^ - 62 - ^2)2 

which may be reduced to the form 



Vc^ 



4(a6 + cd)(ac + bd)(ad 4- 6c) 



(a + 6 + c - d) (a + 6 + cZ - c) (a + c + cZ - 6) (6 + c + fZ - a) 



244 TABLE BOOK AND TEST PBOBLEMS. 

293. First Solution. 

Multiplying by 4 x, 4 x* — 12 x^ + 16 x = 0. 

Extracting square root, (2 x^ — 3 x)"^ — 9 x'^ -\- 16 x == 0. 
Adding to each member x^ — 4 x, 

(2x2 - 3x)2 - 4(2x2 - 3x)= x2 - 4x. 
Completing square, and extracting square root, 

(2 x2 - 3 x) - 2 = X - 2, 
whence x = 2. 

Second Solution, » 

Multiplying by x, x* — 3 x-^ + 4 x = 0. 

Subtracting from this 3 x the given equation, 

x*-9x2 = -4x- 12. 

Adding x2 + 16 to each member, x* — 8 x2 + 16 = x^ — 4 x + 4. 
Extracting square root, x2 — 4 = x — 2, 

whence x = 2. 

294. Multiplying by 4 x, 4 x* — 24 x2 + 16 x = 0. 
Extracting square root, (2 x^ — 6)2 + 16 x — 36 = 0, 

or (2x2-6)2-36=-16x. 

Adding to each member 16 x^ — 12, 

(2x2-6)2+ 8(2x2-6)= 16x2- 16x- 12, 
Completing square, and extracting square root, 

(2x2-6) +4 = 4x-2, 
whence x = 2. 

295. Multiplying by 4 x, 

4 x* - 32 x^ + 76 x2 - 48 X = 0. 
Extracting square root, 

(2x2-8x)2+ 12x2-48x = 0. 
Factoring, etc., (2 x2 - 8 x)2 + 6(2 x2 - 8 x) + 9 = 9, 
whence 2 x2 — 8 x = or — 6. 

x2 — 4x = or — 3; whence x = 4, 3, or 1. 



MISCELLANEO US SOL UTIONS. 



245 



MISCELLANEOUS SOLUTIONS. 



APPLICATIONS OF ALGEBRA. 



296. First Solution. Let ABCD be the rectangle, and E its center. 
Let X = its length, AB ; and y = its width, AC. Then x -\- y = 70, (1). 
In the triangle AIE^ AE = 25, AI = | x. 



and IE 

Hence 



?/, / being a 



right angle. 




+ (^¥=25-2, or 1 + 1 = 625, (2). 

Clearing, x^ + 2/^ = 2500, (3). From 

(1), y = 70 — X. Putting this value of 

y in (3), we obtam, by collecting terms, 

etc., x'^ - 70x = — 1200, (4). Completing square, x- — TOx -f 1225 = 25. 

Extracting square root, x — 35 = i 5, and x == 35 ± 5 = 40 or 30. Hence 

y = SO or 40. The area z^ 40 x 30 = 1200 square feet. 

Seco7id Solution. AB + BD = 70. Let x = AB. Then BD = 70 - x. 
AE=2o = ED. Hence AD = 50. In triangle ABD we have xH (70 -x)2 
= 50^, whence x- — 70 x = — 1200, the same equation as (4) above. 

297. Let X = the base, and y = the perpendicular. Then Vx'-^ + y'^ = the 
hypothenuse. By the problem, x — y = a^ (1) ; and Vx- 4- i/^ — x = 6, (2). 
Adding (1) and (2), Vx"^ + 2/^ — ?/ = a + 0, (3). Putting a + & r= c, trans- 
posing, and squaring (3), x"^ + 2/^ = c- -\- 2 cy -\- y'^, (4). From (1)^, 
x^-^y^ = a- -\-2xy, (5). Subtracting (5) from (4), c^ - a^ + 2 ?/ (c - x) 
-f 2/'^ = 0, (6). From (1), x = a + i/. Putting this value in (6), we have 
c2 — a"^ + 2 2/{c — (a 4- 2/)} +2/^ = 0. Transposing, etc., y'^ -\- 2y(a — c) 
= c2 — a\ Completing square, y'^ -\-2y(a — c) + (a — c)^= (« — c)^ + c-— a-. 
Whence y = c • 
c±\/2c^ 



+ 2/ 



a it ■n/2 cf^ — 2ac 
2 ac = c ± V2 c (c 



(c — g) ± \/2 c (c 



a). Then x 



298. Let ABCD represent the floor, 
and EF= GH= 3. Let x = AH, 
and y = AG. In the triangle HAG, 
x^ + 2/'^ =: 9, whence x := V9 — y'^, (1). 
The triangles AHG and (tDE' being 
similar, x : y : : b — y : a — x, whence 
x^ — ax = 2/^ — ^y- Substituting value 



Put DC=a = 40, BC=b = lS, 




Ah 



of X from (1), transposing, squaring, etc., we get, after dividing by 4, 



246 



TABLE BOOK AND TEST PROBLEMS. 



yi _ 13?/5 + 433.25?/2 + 58.5?/ 
ner's method, we lincl ?/ = 2.81 



3579.75 = 0. By double position, or Hor- 
Then x = VO^^ = .84. Hence 



EG = \/{(40 - .84)-^ + (13 - 2.88)^} = 40.4+ feet. 



299. Let X = number of acres = number of rails. Then 160 x = number 

of square rods, ^ = radius = 4a/ — -, and 8a/ — - = diameter ; also 

8 tta/ — - — 8 VlO -nx — circumference. Since the boards are 1 rod long, 

and the fence is 5 boards high, - = number of rods in circumference. 

^ 5 

Hence - = 8 VlO^, whence x = 16,000 tt = 50,265.5. 
5 

300. Let X — length (in feet) of the inside. Then x -^ \ = outside 
length, and 6 (x + \y^ = entire outer surface, (1). Since the outside sur- 
face includes several edges, it is greater than 49f . The gain is 4 edges of 
the inside length, and 8 edges of the outside length ; that is, 4 x + 8(x + J) 
feet in length; and this multiphed by ^ (or IJ inches) gives the area 
of the said edges r= |x + }. Then 6(x + i)'^ — (|x + })= 49f, whence 
X = "Y = 2| feet, and x + J == 3 feet = 36 inches, the required length. 



301. Let AB = the shorter, and CD = the longer, of the poles ; CA = 
the plane; E = the point of intersection. We then have BC = 40, DA 

= 60, and EF = 15. By similar triangles, 
wehave^£':^i^::^Z>:^C, BEiAF :\bC 
-.AC] whence BE: AE:: BC: AD, and like- 
wise DE': CE::AD:BC. But AD:BC::Z 
:2: hence 7) £7: a^':: 3: 2. Let2x = £'(7, 
3x =: DE, BE = iO -2x, and AE = QO - 
3x. By similar triangles, we have AE: EF 
: : AD : DC, whence, substituting values, DC 
900 300 




60 - 3 X 20 - 
BA, whence BA 



also CE : EF: : BC : 



300 



AD^-CD'-=AC\ 



(1) ; and BC^ - AB'^ = AC\ (2). 

(77)2 = BC^- AB\ Substituting values, OO"^ - 



Hence, equating (1) and (2), AD^ — 

_30(LY^402_/3ooy^ 

20 -x/ \ X I 

Reducing this, we get x* - 40x'^ + 400 x^ - 1800 x + 18,000 = 0, whence, 
by Horner's method, x = 13.954 + . Hence 300^x = 21.5 feet, the shorter 
pole. Then 300 - (20 - x) = 49.62+ feet, the longer pole. 



MISCELLANEOUS SOLUTIONS. 



2i7 



302. Let ABC = the triangle, ED = the parallel ; and make BC =: a, 
AC=b, AB = c, CD = x, and BE 
= y. Then AD = b — x, AE = c — y, 
and DE =z x — y. By similar trian- 
gles, b — X: c — y : :b: c, whence ex 
= i^l/y (^) I ^Iso a:X — y: :b: b — X, 
whence bx — by -{- ax = ab, (2). From 
(1) and (2), 

ac _. ab 




y 



X = 



a -\- b — c 



and X ~- y 



a -\- b — c 



B 



303. Let AB = pole, AD = the hillside, C — the point where the break 

occurs. BC = DC = a. AC=EC=x. AD = 
b, ED — c. CF being a perpendicular, and 

b-c 




ACE isosceles, AF = FE 
& + c 

2 



2 



and FD = 



CE^ = CD^ - FD^ + EF\ Substi- 



tuting values, we have x'^ = a^ — ( 7" 1 + 



(¥) 



Clearing, transposing, and reduc- 



ing, we have x'^ 
Then a -V ^ a'- - 
pole. 



= a-— 6c, and x = Va^ — 5c. 

6c = the original height of 

(From Dr. I. J. Wireback.) 



{x + ?/) (a; 



304. First Solution. Let x + ?/ = ^^, 
X — y = BC. Now, the product of the 
sides divided by their sum is equal to the 
side of the inscribed square. Therefore 

also 



-^ = 12, or x2 - ?/ = 24x, (1) 



(x + y) -^(x-ij) 

AB^ + BC^ = (X + ?/)2 + (x - ?/)2 = 352, or 2 x^ +2?/-^ 
= 1225, (2). Adding 2^x (1) to (2), we have 4x2 
— 48 X = 1226. Completing square, 4 x'^ — 48 x + 
144 = 1369, whence x = 24}. Putting this value of 
X in (2), we find y = 3]-. Then x + ?/ = 24} + 3} = 
28, and x-y = 21. 

Note. — The truth of the statement made in the beginning 
of the foregoing solution may be seen by examining Equation 
(2) in the following solution of the same problem. 

Second Solution. (See figure in above solution.) Let AB = x, BC — y. 
Then x^ + 1/2 = 1225, (1). By similar triangles, AB : BC : : AE : ED, or 




248 TABLE BOOK AND TEST PROBLEMS, 

x:y : \x — 12:12 ) whence xy = 12(x + ?/), or ^^:=: x + ?/, (2). Squaring 

i z 

(2), ^ = x2 + 2x2/ + 2/^. Transposing, ^ - 2x?/ = x^ + 2/^, (3). 
Taking the value of x- + 2/^ from (1), we liave —^ 2x2/ = 1225, o^ 

xV - 288 xy = 176,400. Completing the square, xV - 288 xy + 20,736 
= 197,136, whence xy = 588, (4). Putting this value of xy in (2), we 

have := X + 2/ = 49. Then x = 49 — 2/, and (4) becomes (49 — y)y 

= 588, or 2/2 - 49 2/ = -588, whence 2/ = 21 or 28. Hence x = 28 or 21. 

305. Let X = number of acres = number of boards. Then 160 x = 
number of square rods. - = number of boards on one side. There are 

^ X / X 

14 boards in 1 rod. Hence — = number of rods square, and — 

56 V-56, 

160 X. x^ = 501,760 X, and x = 501,760. (From Dr. I. J. Wireback.) 

306. Volume of globe is 12^ x .5236 = 12^ x Jtt. Put Z) = inside 
diameter of hollow sphere. Then D -{- \ = the outside diameter. {D -\- \Y 
X 1^ TT = space occupied by hollow sphere. D^ x J- tt = space or volume 

inside the shell. Then \ir{D + \Y - JttD^ ^ i23 x Jtt. Dividing by 
^TT, we have (Z) + lY - D^ = 123, or D^ + J D^ + y's ^ + eV - -^^ == 1^28. 
Clearing, etc., 48Z>2 + 12Z> = 110,591, or 16 D^ + 4i; = 36,863§. Complet- 
ing square, 16 Z>- + 4 Z) + J = 36,863| + J, whence, extracting square 
root, etc., D = 47.87 + , and D -\- i = 48.12+ inches. 

307. Let X = altitude. Then 2x = base, each of the equal sides = 
xV2, and the area = x^. The area -f- perimeter = J radius of inscribed 

circle. Hence radius = 2x2 -^ (2x + 2xV2) = — Then — ~ — ^ x 

1 + \/2 1 + V2 

r 1 ■\/2 — 1 /- 
= - =z -. Rationalizing the denominator, we have = v2 — 1. 

^ 1 + V2 - 2-1 

308. Let ^^0=the triangle, ^.B = 2x, AC=Sx, BC=b. On BC 

let fall the perpendicular AD, and 
let 2/ = BD. Then CD = b - y. 
In right triangle ADC, AD'^ = 9x2 
-b'^-\-2by- 2/2, (1). In right 
triangle ADB, AD'^ = 4 .x2 _ 2/2, 
(2). Equating (I) and (2), we 
have 9x2- 62+ 2 by-y^=4:x'^- 2/2, 
or 5 x2 + 2 by = b'^, (3). The area 

of ABC = h (BC X AD) = a. 

Hence AD = ^ ^, and AD^ = — . Substituting in (2), we have 4x2 - 
b b^ o V y, 




MISCELLANEOUS SOLUTIONS. 



249 



A 1 

y2 — _3-^ (4) . Comparing (3) and (4) , and eliminating x, we get 5 if- 



h^ 



20 a^ 



46 



100 a2)-^^. Putting this 
5 



+ 8 6?/ = 4 &2 _ f^^L^, whence y = IbVJWb^ 
value of y in (3), and solving, we have 

and AB = 2x = twice this ; and AC =Sx = three times this. 



100^2 



309. First Solution. Denote the angle A by 6. Then the angle ECD 



= 90 - 6>, the angle EDB = 2 0. 
and the angle BEA = 90-6/. 
Hence the triangle ADE is 
isosceles, and AD = BE = 80. 
Now, by similar triangles, 
AB : BE : : BE : BC. Let 
X = BD. Then BE^ = (80 + x) 
(20 + x) =: x2 + lOOic + 1600, 
(1). In right triangle EBD, 
BE-^ = DE^- BD'-=6i00 - x% 
(2). Equating (1) and (2), 
aj2 + 100 X + 1600 = 6400 - x\ 
= 5500, and BE = 10 VU. 



The angle DEA = 6, the angle BEC = 6, 




whence x = 30. Then j^^^ ^ 80^ - 30--2 



Second Solution. The triangle ADE is isosceles, hence DE = 80. 
Put DB = x, EB = y. x"^ -^ i/ = DE^ = 6^00, (1). The angle AEB 
= 90 ~ ^ = the angle ECD, and the angle BEC = 6> = the angle A : 
hence the triangles AEB and ^(7^ are similar, and we have sinAiy 
: : sin AEB : x + 80, and sin ^(or BEC): x + 20 : : sin ^£'J5(or ECD) : y ; 
whence ?/ : :*^ + 80 ; : x + 20 : ?/, or ^/'^ =r x^ + 100 x + 1600, (2). Substitut- 
ing in (1), we have x^ + x^ + 100 x + 1600 = 6400 ; whence x = 30, and x^ 
= 900. Using this value of x^ in (1), we have if = 5500, y = 74.16+ = EB. 



310. First Solution. (See first figure on next page.) Let AB = 74 
feet, height of tree ; BD = 34, the hillside ; BE = 18, the horizontal dis- 
tance ; BC = x; DE = y. Then 

2/ = 74 - X — Vx^ + 324 (1) 

The triangle BED = triangle BCD — triangle BCE. Applying the 
rule for finding the area when the three sides are given, we have 



V 



(b2-^-y2)(y2_ 16-2) 
16 



= V1U80(54 -x)(x- 20)- 9x 



(2) 



Prom (1) and (2), we find x = 24, and y = 20. 



260 



TABLE BOOK AND TEST PROBLEMS, 



Second Solution. Let BC = x, BF = y, FD = z, BFD being a right 
angle. Then we have (x + y)'^ + 34- = (74 — x)'^ + ?/"^, or .74x + x?/ 

= 2160,(1). 2/- + ^-^r=:34-2, (2). x:l^:'.x-\-y:z, 
^ or 18x + 18?/ = x^, (3). From (1), y = 

2160- 74 x ^ Substituting in (3), 18 x^ - 1332 x 

X 

+ 38,880 = x% (4). From (4), 

_ 18(x^- 74x4-2160) 

X- 

Substituting values of y and z in (2), and 
reducing, we have 1161 x^ - 91,908x5 + 1,959,- 
876x^- 25,894,080x +377,913,600 = 0. Solving, 
Q X = 24: =z height of stump. 



311. Let ABCD represent the square, Pthe 
point within, PA = a, PB = h, and PD = c. 
Draw EF perpendicular to CD. Put AB = BD 
= X, AE = ?/, EP = z. Then BE = x - y, and 
PjP = X — z. By the Pythagorean theorem, we 
have AE^ + J5;p2 = aP'\ or ?/2 + z"^ = a'-, (1). 
^iS2 + EP-^ = BP^ or (x - ?/)2 + ^2 ^ ^2^ (2). 
FD^ (or ^P^) + PP2 ^ px^-2^ or (x - 2/)2 + 
(x-^)2 = c^ (3). Subtracting (2) from (3) 
gives x2 — 2x^ = c- — b'^, (4). Subtracting (1) 
from (2) gives x'^ — 2xy = b^ — d^, (5). Trans- 
posing, etc., 2 xy = x:^- — {W — a^-) , (6). From 
(1), 2J = Va2 — 2/2. Putting this in (4), we 
have 





x2-2xVa^-2/^ = c2-62^ 
2 xVa^ - y^ = x^ - (c^ - b'^) 



(^) 



or zxva^ — y- = X- 

Adding (6)2 and (7)2, we have 

4 a2x2 =(^2 - (62 _ a2)}2 + {x2 - (c^ - b'^)} (8) 

For convenience put b'^ — ^2 — ^^^-2^ and c2 — b- = ?i2. Then (8) 
becomes 4 a V =(x2 — ^2^2 _^(^^2 _ ^-2)2^ (^9^^ Expanding (9) and col- 



lectinjr terms, 



:* - x2(m2 + 7i2 + 2 a2) = - V^lAlUl.^ (lO). 



Putting 711^ + 
g. Com- 



n2 + 2 a2 = 2_p, and ^^ "^ ^^^ =(7, (10) becomes x^ - 2;\x2 : 

pleting square, x* — 2^9x2 -f ^2 _^2 _ ^^ whence x2 = j) ± \/p2 _ q. Then 

X = Vp i \/|)2 _ (7. 



MISCELLANEOUS SOLUTIONS. 



251 



312. Let ABC (Fig. 1) be the triangle; AD, CG, and BE, the poles ; 
and H, K, and /, the points on the sides which are equally distant from 
the tops of the poles between which they lie. Then AD = 50, BE = 40, 




>i^ 




^ M 



Ca = SO, AB = BC = AC =200. J.et AH = a, BH=h, and DH = EH 
= X. Then x^ = 50^ + a" = 40^ + b'^, whence b'^ - a^ = 900, (1). BH + 
AH =b + a = 200, (2). Dividing (1) by (2), 6 - a = 4.5, (3). Adding 
(2) and (3), 2b = 204.5, and b = 102.25 = BH. Then a = 200 - b = 
97.75 = AH. In a similar manner we find AK= 96, CK= 104, BI = 
9S.2d, and CI= 101.75. Perpendiculars erected at the points H, 7, and 
K, will meet in a point, which is the required point, being equally distant 
from D, G, and E, the tops of the poles. In Eig. 2 erect perpendiculars 
at H, I, and K. Let be the point at which they intersect. Produce 
10 to M in AB, . Bisect BC at F, and join AF, AO, BO, and CO. The 
triangle ABC being equilateral, AF and 311 are parallel, and therefore 
AM = 2 IF. BF- BI= IF =100 - 98.25 = 1.75. Hence AM= 1.75 x 2 
= 3.5. MH = AH - AM = 97.75 - 3.5 = 94.25. The triangles ABF 
and MOH are similar. Therefore MO = 2 OH. Let 0^ = x. Then J/0 
= 2x. MH^ = MO'- - Om = 4 x^ - s:^ = 3 ^2. But 3/ir = 94. 25. Hence 
3 x^ = 94.25^ an d 0^ = 54.415+= OF. In the triangle AOH, AO = 
\/{AH^ 4- OH^) = V (97.75-2 + 54.415-) = 111.875 feet, the distance the 
fourth pole must be placed from the foot of the 50- foot pole. In the tri- 
angle BOH, BO = V(BH^ + OH^) = 115.82 7 feet, the dist ance from the 
40-foot pole. In the triangle AOIu OK = V iAO' - AK^)= 57.428 feet. 
In the triangle COA", CO = V^CK'^ + OTC^) = 118.811 feet, the distance 
from the 30-foot pole, 
fourth pole. 



Then VllS.Sil^ + 30"^ = 122.53-1- 



length of 



252 



TABLE BOOK AND TEST PROBLEMS. 



313. Let MBN be any quadrant, A the centre of its inscribed circle, 

and F tlie centre of the required circle. 
Through A and F draw the radii of the 
quadrant, BAG and BFH. Let fall AD 
and FK each at right angles to BN^ and 
draw FE at right angles to AD. Pat 
B = BG or ^.V, r =: ^i> =: i?( V2 - 1) := 
.4142135624 7^; and let x = FK, the radius 
of the required circle. Then BF = R — x, 
AB = R - r, AF = r -\- x , AE = r - x, 
and FE = BK - BD = ^{{R - x)'^ - z^} 
- V{(i? - r)-^ - r% (1). Also FE^ = FA^ 
AE^ — (r + x)'^ — (r — x)^, (2). Equating values of FE^, we have 

{V[(i^^x)--^-x-^] - V[(R-ry-r^]}'2 = (r + x)*^ - (r - x)2 (3) 
Simplifying (3), {\/(R^ - 2 Rx) - V{R^ - 2 i^r)}2 = 4 rx (4) 

Expanding (4), and dividing by 2, 

- V{/^4 _ 2(r + x)is;3 _)_ 4 R-2 ^^J :::, 2 rx - jK2 + (r + x)i? (5) 
Squaring (5), expanding, uniting like terms, etc., we have 

(4 Rr + 4 r2 + R')x'^ - (6 Rh' - 4 i?r2)x = - R^r^ 
3 i?^r - 2 i?r2 - 2 i^?- V(2i^ 
4 /Vr + 4 r- + i?2 




4:Rr) 



(6) 
(7) 



From (6), x = 

But r = R(^V2 — 1); and substituting this in (7), we have 

^ ^ 3 /^3( V2 - 1) - 2 i?^(3 - :. V2) - 2 i?3(3 V2 - 4) 
4 i?-^( V2 -1)4-4 R\3 - 2 V'2) + /^^ 
^ 7?(V2-1) ^ ig(5V2 - 1) ^ .1238993431 R. 
9-4V2 49 

Hence the radius of the required circle in any quadrant is found by mul 
tiplying the radius of the quadrant by the constant decimal .1238993431. 



GEOMETRICAL DEMONSTRATIONS, ETC. 
314. Separate iV^into any two factors, m and n. Then mn = N, 



With 




center O, and radius equal to I {m + n), describe 
a circle ABD. On the diameter CD take C/ 
equal to the smaller factor, and through 7, at 
right angles to CZ>, draw the chord AB. Then 
A I or IB is the square root required. Since the 
chords AB and CD intersect each other, we 
have, by geometry, AI x IB = CI x ID. But 
by construction AI = IB. Therefore AI^ or 
IB" = CI X ID = mn. Extracting the square 
root, ^4/ or IB — \/mn = VX q.e.d. 



MISCELLANEOUS SOLUTIONS, 



263 



315. (See cut in preceding solution.) Take m = 3. Then ^ = 15. 
Describe a circle whose radius is ^(3 + 15), or 9 units, and take CI = 3. 
Then AI = IB ^ V45. 

316. As 19 is a prime number, we must take m == 1, and n = 19. 
Describe a circle with a radius == ^(1 + 19) =: 10 units, and take CI=1 
(see cut in Solution 314). Then AI= IB = Vi9. 

317. First diameter = D = first term of series. 

The seventh diameter = — • Hence the ratio is expressed thus : D: — 

8 8 

This we see is 8. 



318. The first term is Z>, the ratio , and the last term 0. In geo- 
metrical progression, s ^ ~ ^ o„-u„^,-x„4-,- .,„ t 



r 


- 1 










1 

V2 


D 

-1 


1- 


D 




1 

V2 



319. Let abed and NAFO be the two 

inscribed squares, and C the center of 
the circle. By geometry, 

be^ = Ne x eP= CP'^ - Ce\ 

But Ce^ = i be"-. 

Therefore 

5e2 :^ (7p2 _ 1 ^g2^ or CP^ = f be^. 
OP'" =2 CP2. 

Hence OP-^ = f be^, or 2 OP-^ = 5 be^. 
Therefore be^ : OP^ : : 2 : 5. q.e.d. 




320. Let AEBD and ghkc be the in- 
scribed squares, and c the center of the 
circle. By Pythagorean theorem we have 
/ic2 =r 2 cJc-, and cD'^ = BE^ -\- cE\ But 
he = Be. Therefore 2 ck'^ = BE^ + cE^ 



= BE-^-^ \BE^ 



\BE\ 



or 8 c^•2 = 



5 BE^ ; that is, BE^ : ck^ : : 8 : 5. q.e.d. 




A c E k 



254 



TABLE BOOK AND TEST PROBLEMS, 



321. Let ABC be the inscribed equilateral triangle, and the center 

of the circle. Draw CD at right angles to 
^ AB^ and join AG, We are to prove that 

CA: 00 (or J.0) : : V3 : 1. Since by similar 
triangles AC.ADi.AO.OD, and AC =2 AD, 
therefore A0=^2 0D. Then DC = SOD. 
AD = VC^O- - OD-^) = ODVS. Then AC 
= 2 ODVS, and OC^ 0A = 2 0D. There- 
fore AC : (70 (or AO) : : 2 0X> V;J :2 0D :: 
V3:l. 

322. The area of the triangle is 61.48 + 
square feet. But the area of a triangle is 
equal to its perimeter multiplied by half the radius of its inscribed circle. 
Hence 61.48 = 36 x Jr = 18r. Whence r = 3.415 feet. 




.E 



a.-'' 



323. Let ABCD be the board. Then ^0 == 10 feet, and AB = 2 feet. 
Take BI = 1 foot, and cut oft' BAL Take AG and IK each = 5 feet, 
and cut off the two pieces AIG and 
IGK. We now have the four pieces. 
Since AI = V5, we have BI: AI: : AI: 
AG. Hence, the angles AIB and GAI 
being equal, the triangles are similar, 
and AIG is a right angle = IGK. To 
form a square, let AIG and GKI occupy 
the positions KHF and ECG^ and ABI 
the corner CDF. EFGH is the square. 
GH=2Vl. 




H 



GK= KH = AI=Vl, and 



EG = GI= V{AG-' - AI^) = \/20 = 2 VS. 



324. Let AB be a diameter, and CD any chord of the circle. OP = the 

distance from the center to any point, P, of the 
chord. 

BP = OP+OB, 

and AP = AO or OB - OP. 

Hence, multiplying, 

BP X AP =: OB^ - OP^. 

But, by geometry, 

BPx AP = CPx PD. 

Therefore OB^ - OP^ = CP X PD. Transposing, OB^ = OP^ -\- CP 

X PD. Q.E.D. 




MISCELLANEOUS SOLUTIONS. 



255 



325. Let MNA be a vertical section of vessel, EEC the water not 
frozen, and OH the thickness of the ice. Volume of hemisphere = ^2 ^-^^ 
= 7.0686 cubic feet. The water not 
frozen forms a segment of a sphere 
whose height is HC = AO -(OH -^ AC) 

— 18 _ 10 = 8 inches = h ; and the radius 
of the sphere is OC = AO - AC = 18 

— 5 = 13 inches = B. The volume of a 
spherical segment of one base, as EEC, 
is Trh\B — ih). Substituting in this 
formula the values of h and i?, we find 
the volume of the segment to be 1.20234 
cubic feet. Then 7.0686 - 1.20234 = 
5.86626 cubic feet, volume of ice. 





326. Let EBG be a circle tangent 
to major axis AA' at focus E^ and 
passing through i?, the extremity of 
the minor axis. Since the circle is 
tangent at focus E, its center will lie 
in the latus rectum produced. Join 
EB and GB. The triangles EBG 
and ECB are similar, and therefore 
BC : EB : : EB : EG. But EB = AC. 
Hence BC:AC::AC: EG. q.e.d. 



327. Eirst Solution. Let EEIH be the circle; ABCD the circum- 



scribed, and lEEH the inscribed, square 
circle. The difference between the squares 
is the four equal triangles, EAI, IBH^ 
HDE, and ECE. Hence IBH = J of 72 
= 18 square feet. Let x = IB: then area 

IBII=xx- = — = IS, whence x2 = 36, 

2 2 



and EH the diameter of the 



and x = 6. AB = IB -^ AI 

EH = diameter required. 



2 a: = 12 = 



Second Solution. The equal triangles 
EAI, ECE, HDE, and IBH are each J 
of a square, and combined are equivalent 
to two equal squares, the side being 
AI= AE = EC, etc. The area of the four triangles = 72 feet. 




the area of each of the two squares is 36 feet. 
6 feet = AI: and AB or EH= 12 feet. 



Hence 
Therefore the side is 



256 



TABLE BOOK AND TEST PROBLEMS. 



Third Solution. By condition, AB^ - EV- - 72, (1). AB — EH, and 
AB'- = EH\ But EH-^ = EP + IH^ = 2 ELK Hence El^ ^ i AB^. 
Substituting in (1), we have AB^ ~ hAB^ — 12, whence ^5- = 144, 
ABz^ 12 = EH. 

Fourth Solution. Assume an isosceles right triangle whose sides are 
1, 1, and •\/2. Its area is J. The assumed triangle is similar to each of 
the four equal triangles, IBH, etc. Hence, by similar triangles, J : 18 : : 
1 : IB^, whence IB^ = 36, IB = 6. Then ^^ = 12 = EH 

Fifth Solution. The side of an inscribed square is to the side of a cir- 
cumscribed square as 1 is to V2. Therefore AB^ : EF^ ; : 2 : 1, or EP = 
1 AB\ But AB^ - El^ = 72. Hence AB^ - i AB^ = 72, or AB^ = 144, 
^^ = 12 = EH 



328. Let ABCD be the parallelogram, AC its diagonal, to be trisected. 

Bisect BC and AD in E and F, 
and join ED and FB. Then will 
BF^ndED trisect AC, and AH 
= HG= GC. Since BE and FD 
are equal and parallel, BE and 
ED are also parallel. The trian- 
gles AFH and ADG are similar, 
and therefore AF.FD: :AH: HG. 
Therefore AH^HG. Similarly CE-. 




But by construction AF = FD. 
EB-.'.QG: GH. But CE = EB. 

Q.E.D. 



Therefore GC = GH. Therefore, by 



axiom, AH= HG^GC. 



329. Let ABCD be the lot, ^ its center, EG = 50 feet, EK = 40 feet. 




Then GK 



40-' = 30 feet. 



The horse can graze over the rect- 
angle EG HI, and over the two 
equal segments GI and FH. FK 
= KG = 30 feet, and EG = 60 
feet. Hence the rectangle is 60 
X 80 = 4800 square feet. FH = 
GI = 80 feet = base of segment, 
C li I ^ and the height is EM - GK = 50 

— 30 = 20 feet. The area of one segment is therefore ^(80x20) + 
(203 ^ 1(30) - 11163 square feet. Then both segments = 2233 J feet, which, 
added to the rectangle (4800 square feet), gives 7033 A square feet, the 
area grazed over. 



MISCELLANEOUS SOLUTIONS. 



257 



330. Let A^ B. and C be the centers of the circular fields. Put the 



area of each field = 80 x 160 = a square rods, 
and "Ir^ AB = AC= BC, the sides of 
the equilateral triangle ABC. The area 
of the triangle ABC =r'^\^'^. But this 
includes three sectors, each of which = \ 
of a circle, since its angle is 60°. The 
three sectors =: J of a circle = J a. Hence 
the area inclosed hy the circles must be 
r^ V8 — la; or, substituting the value of 

r = V-, we have VH — A a. Substitut- 

ing values as given in the special prob- 
lem, we find the inclosed area = 656.78 
square rods. 



Then radius 



A' 







331. Let a = area of each field. The area of a circle — Trr^ = a. Then 



1 ^ _ TT?- 

^ 2 



Substituting this in the formula for inclosed area as given 



above, we have r'^VS — J irr^ = 1 acre = 160 rods, or (Vs — J 7r)r^ = 160, 
whence r = 31.5+ rods, and diameter = 2r = 6S-\- rods. 



332. Let ABCD be the rhombus, and 
AD = 24, inches. Draw BOC^ the other di- 
agonal. BO is perpendicular to AD^ because 
the diagonals of a rhombus intersect at right 
angles. The area of the triangle ABD =-- 
i ABCD = 10S inches. 108-12 (half the 
base) =9= OB. AO'- + OB'- = AB^; or 122 
4- 9'2 = AB^^ and AB = 15. Hence the perim- 
eter (four sides) = 60 inches. 




333. Erom the center of larger ball to either wall is 12 = i? inches. 
Hence from its center to the corner is ^/SB^ = BVS. Subtracting from 
this the radius of the ball, we have BVS — B = B(VS — 1)^ the distance 
from the corner to the surface of the ball (in the direction of its center). 
Let r = radius of smaller ball. Then from its center to corner is rV3, 
and rV3 + r == r(\/3 + 1) = distance -from corner to surface of larger 
ball. But this was found above to be i?(\/3 — 1). Therefore r(\/3 + 1) 
= B(VS - 1). But E = 12, and VS = 1.732. Hence 2.732 r = .1S2B 
= 8.784, whence r = 3.215 + , and the diameter is 2 r = 6.43 inches. 
ellwood's test prob. — 17. 



258 



TABLE BOOK AND TEST PROBLEMS. 



334. Let AHFE represent a vertical section 
through the earth's center, P the position of 
the person, and AB the base of the zone to be 
seen. By geometry, the surface of hemisphere 
is to the surface of zone as HC is to DH, or 
t} : J : : r : DH, whence, assuming radius of earth 
= 4000, i)7f= 26661 miles. By similar tri- 
angles, DC: BC: : BC: CP. But DC=HC 
-Diy= 4000 -26661 = 1333 J. Hence the 
preceding proportion becomes 1333 A : 4000 : : 
4000 :(7P, or 12,000. CP - Ci7 = 12,000 - 
4000 = 8000 miles = HP, the required dis- 
tance, which is the same as the earth's 
diameter. 



335. Let J.PCbe the given triangle, Pthe 
point witnin, CP = a, AP = b, and PB = c. On PB 
construct the equilateral triangle PBD, and join 
CD. In the triangles CBD and APB, DB = BP, 
AB = BC, and the angle OPZ> = the angle PBA, 
each being equal to 60" — the angle PBC. Hence 
CD=AP=b. We now have PC = a, CD = &, and 
PD = c, to find the angle CPD = 0. Then the angle 
GPB = 60° + d, and we have two sides (CP and PB) 
and included angle to find PC, the required side. 





336. Let be the center of the circle, OC = 0E = S, the radius; 

= 4; CD = 6; AF = 2 ; C/ = 2.5. 

07= jVn = 1.658 + , 

FO=VZ = 2.236, 

FE=S-Vd = .764, 

^/=:3- jVn = 1.342. 

Area of segment AEB, whose height is 
EF = .764, and base is AB = 4, is found by 
rule to be 2.093 + . Area of segment CED, 
whose height is EI = 1 .342, and base is CD 
= 5, we find by rule to be 4.715! Then 4.715 - 2.093 = 2.622, the re- 
quired area. 




MISCELLANEO US SOL UTIONS. 



259 



337. Let be the center of the sphere, OB = 2, AB = 2, the diameter 
of the auger. The volume removed is a cy Under, ABCD^ and two seg- 
ments, AEB and CD I. FE — height, and 
AB = base of segment. OB^ - BF^ = F0\ 
or 4-1 = 3. Hence OF = VS = 1.732. 
EF = OF - OF =:2 - 1.732 = .268. The 
volume of a segment may be found from the 
formula ^ '7rh(Ji^ + 3 r-). Using this, we find 
the segment AEB — .431 of a cubic inch, and 
both segments = .862 of a cubic inch. The 
length of the cylinder = diameter of sphere 
less the height of the two segments, or 4 — 
.536 = 3.464 inches. Area of its base = ttv'^ 
= 3.1416. Volume of cylinder = irr-h = 3.1416 x 3.464 




Then 



10.8825. 
10.8825 + .862 = 11.7445 cubic inches, the amount bored away. 

338. Let ABCD be the semicircle, E its center, and F and / the centers 
of the two given circles. Then 

FH = a, IK = &, 

and ED = EC = r, EF= r - a, 

EI=r-b, FI^a-\-b, 

EH = y/{r-ay^-a- 



EK = V (r 



2 ar. 




Hence {EH-^ EKY + PG"^ = FP ; and, since FG = FH- IK, we have 

{ Vr-2 - 2 «r + Vr2 - 2 ftrf + (a - 5)2 = (a + by\ 
or 2 r2 - 2 r(a + 6) + 2 V{(r2 - 2 ar) (r- -2br)} = 4t ab, 
or r2 - r(a + 6) + r\/{r2 - 2(a + 6)/- + 4 a6} = 2 a6. 
Transposing, squaring, and reducing, we get 



+ 



r^ 






6_ 

ab 



1^ 

62' 



whence 



H 



-ifi + l),and,- = 



2Va 6 



V5i 






1 
ab 2 

339. Let F = the number of cubic inches of copper, t = the thickness 
of the shell in inches, B = the outer radius, and r = the inner radius. 
Then, by the problem, B - r = t, (1). |7r(i?3 _ ^3^) ^ |r ^2). By an easy 
process of elimination, we have 



='{VVM-}-^-=»{a/|H-} 



260 



TABLE BOOK AND TEST PB0BLEM8. 



Hence the capacity of the kettle in cubic inches is expressed by 



irr^ 






340. First Solution. On the hypothenuse of the right triangle BCE 
(Fig. 1) draw the half square BE A. Erom A let fall AD, a perpendicular 
on CF produced. The triangles BCF and ADE are equal, BC= DE, 
and EG = AD. The area of the quadrilateral ABCD is measured by CD 
multiplied by half the sum of the parallel sides AD and EC. Hence 
^resi, ABCD =CDxi(SG + AD) = i (EC ^ BC)'^. But area ABCD = 
area BEA-{-BCE+ADE=i(BEy+(iECxBC). Therefore {EC+BC^ 
= BE-^-{-2(iECxBC), or EC'^ + BC^-{-2{ECxBC)=BE-^-\-2{ECxBC). 
Therefore EC'^ + j^C^ =: BE'^. Q.e.d. (From Mathematical Magazine.) 

Note. — This is known as the Pythagorean Theorem, and the demonstration taken 
from the *' Mathematical Magazine " is tho one given by President Grarfield. 





Fig. 2. 

Second Solution. Let ABC (Fig. 2) be a right-angled triangle. We are 
to prove AC'^ — AB^ + BC^. Let fall on the hypothenuse the perpen- 
dicular BD. Then AB^^AC x AD, and BC'^ =AC x CD. Adding, we 
have AB^ -\-BC'^ = AC x {AD + CD). But AD -^ CD = AC. There- 
fore AB^ + BC'^ =ACxAC = AC-\ q.e.d. 

Third Solution. 



to Z>, making FD 



Let ECF be any right-angled triangle. Produce CF 
: CE. On CD describe a square ; also on EF. CD =: 
CF -f FD, and CD^ = CF'^ + FD^ + 2 CF x 
FD. But FD = CE (by construction). Hence, 
substituting, CD'- = CF'^^ CE'^+2CFxEC, (1). 
Since DF = EC, FK- FE, and Z) is a right 
angle, therefore DK— FC, and the triangle 
FDK = the triangle ECF. Similarly the tri- 
angles HBK and HAE are equal to the triangle 
ECF. The area of the square CDBA is equal 
to the area of the square EFKH + the area of 
the four equal triangles. Now, the area of the 
triangle ECF = J CF x CE, and the area of the four triangles = 2 CF x 




MISCELLANEOUS SOLUTIONS. 



261 




CE. Hence CD^- = EF^ + 2 CF x CE, (2). Equating values of CD"^ in 
(1) and (2), and subtracting equals from each side, we have EF^ = CF'^ 

4- CE'^. Q.E.D. 

Fourth Solution. Let ABC be a triangle, right-angled at B. On AC 
construct the square AC ED. Draw ^(r perpendicular to J. (7, and pro- 
duce BG to K, making BK= AD. Join DK 
and EK., and produce AB to H. We are to 
prove AC- = AB'^ + BC\ ABKD and CBKE 
are parallelograms, each by construction hav- 
ing two opposite sides equal and parallel ; but 
ABKD = the rectangle AGFD, both having the 
same base and altitude. For similar reasons 
CBKE = the rectangle GCEF. Hence the two 
parallelograms are together equivalent to the 
two rectangles which form the square ACED 
= AC'l The triangles ABC and BIIK are 
equal, being mutually equi-angular, and having 
their homologous sides perpendicular each to each, ^(7 being by construc- 
tion equal to BK. Hence BC = BR, and AB = HK. Area CEKB = 
BCx BH= BC^ and the area ABKD = BA x HK = BA^. Since the 
two parallelograms = the two rectangles — AC^, it follows that AC'^ = 

BC^ + BA^ Q.E.D. 

Fifth Solution. J^et ABC loe a triangle, right-angled at B. We are to 
prove AC^ = AB'^ -\- BC^. Construct squares on each side. Draw BI 
parallel to CK, and HB and 
CD. ABC and ABE both 
being right angles, CBE is a 
straight line. 'Likev^ise AB F 
is a straight line. DAC = 
BAH, each being a right 
angle increased by the angle 
BA C. Therefore the triangle 
DAC = the triangle BAH. 
Now, the parallelogram A I 
is double the triangle BAH, 
and the square BD is double 
the triangle DAC. Therefore 
the parallelogram AI = the 
square BD. Similarly, by 
joining BK and AG, it may 
be shown that the parallelo- 
gram C7=the square BG. 




262 



TABLE BOOK AND TEST PROBLEMS. 



But the square on AG = the two parallelograms —AI-]- CI— the square 
BD + the square BG. Therefore AC^ = AB'^ + BG'^. q.e.d. 



341. Let ABD represent a vertical section of the wine-glass, passing 
through the axes NFCKD and FCK of the glass and immersed hall, 

whose center is at C Draw the radius 
CG, G- being the point of tangency of 
the l3all and glass. Let the water line 
HE be tangent to the ball at the point 
F, Put AN^NB=.r^\^ inches, NB 

=. d — ^ inches, and - = — We shall 
n 3 

have, by geometry, V —\ irr-d — vol- 

irr^d 




ume of wine-oflass, and v 



3 71 



vol- 



ume of water in the glass ; also BD 
= V(?-^ + d^). Let X == GG = CF=GK, 
the required radius of the immersed 
ball. Comparing the similar right tri- 
angles BND and GGD, we have NB: 



BDiiGG: GD; that is, r:V(r- + (^^) 
I'.X'.GD. Therefore GD = ^(r!±^^ 



whence FD = CD-{-GF= ^^^"^ "^ ^^^^ + x. By comparing similar sohds 

r 
(geometry), we have (NDy:(FDy: : F: volume of cone HED ; that is, 



d^ 



:V(r'^ + d^) 



+ 



^3 1 

x\ :: -f^d: volume of cone HED. Therefore vol- 
/ 3 

ume of cone HED = — | xV{r^_dr)_ ^ y^ ^^^ .^^ ^^^^ ^^^ volume 

Sd^ I r i 

of the cone HED, we subtract the volume of the immersed ball, the 



remamder must equal v. Whence :r^ i ^ -{- x y ^— = v 



\d^ I 



J 



f :L^ (1). Multiplying (1) by §, ^l^ { ^^^^^I+H +. }'- 4.^= '_1^, (2). 

Or -i-{xV(r^ + c/-) + rx}3- 4 x^ = — , or n {xV(?'^ + c^;+ ?'x}3- 4 nd^rx^ 
d-r n 

= rW, or 7iic3 {\/(r^ + cZ-) + rf- 4 iicfrx^ = r^cZ>, (3). Whence x^ == 
^^^^■^ -, andx^ ^'^ 



71 



{ V(H + fZ-) + rf - 4 ?z(^-V 



V [W{V(7'2 + d-^) + r}3 - 4 ?l(^V] 



,V(« 



13V10 + .5 



185 



= 1.88790741- inches. 



(From B. F. Burleson.) 



MISCELLANEOUS SOLUTIONS. 



263 



TRIGONOMETRICAL SOLUTIONS. 



342. Let be the center of the earth ; OP, OA, 
and OB, radii. By the problem, ah = ^ AB. But 
ab : AB :: oa : OA^ since similar arcs are as their 
radii. Therefore oa = i OA. Let fall the perpen- 
dicular ac. Then Oc = oa = ^ OA = J r. By 
trigonometry, Oa : sin 90° :: Oc : sin Oac, or r : 1 
: : Jr : sin Oac; whence sin Oac — .5, which is the 
sine of 80°. Hence the angle AOa = 90°- 30° = 
60°, the required latitude. 




343. Let a = the angle ACB, r = AC = 5. AD = r sin a: CD 



r cos a. Area of circle = vrr^, area of sector ACB 



860 



7r?'2, area of tri- 




angle A CD = J r^ sin a cos a ; semi-segment 



1 

To 



= T-^n irr- 



Now, the sector — the triangle 



the semi-segment. 



Hence irf^ — 

860 



r- sm a 



cos a = To^^'^- Dividing by Trr^, we get 



1 . 

sm a cos a = 

27r 

cos a = 86. 



10' 



Clearing, a 



180 



360 
sin a 



But sin a cos a = J sin 2 a. Sub- 
stituting this value in the equation, we have 
a - 28.648 sin 2 a = 36. Solving by method of 
approximation, we find a = 59° 55' 25'^ Sin a = .86536. AD = rsina 
= 4.3268 feet = J of the required chord. Hence chord = 8.6536 feet. 

344. Fii^st Solution. Let C be the center of pond, P the point with- 
out, and A and B the points of tangency. Since the angle APB is 60°, 
the angle ^PC is 30°. PAC being a right angle, the 
angle ACP = 60'^. Then, by trigonometry, sin 60°: P 

18 : : sin 30° : AC, whence AC = 10.39 = radius, and 
20.78 = diameter. 

Second Solution. The angle P being 60°, and AP A 
and BP being equal, the triangle ABP is equilateral, 
and AB = 18. PC bisects AB in E, and EA = 9. 
The triangles ^^P and ACP, being right-angled, and 
having the common angle APC, are similar. Hence 
EP:AP::AE:AC. Now, ^P = Vl8-^ - 9^ = 9V3. 
Then 9 V3 : 18 : : 9 : J.0, whence AC = 10. S9 ; and AD, the diameter 
20.78 rods. 




264 



TABLE BOOK AND TEST PBOBLEMS. 



345. Area of wheel = Trr^ = 4 tt. Let ABD = sefirment in mud. Then 



ED = 1 foot, CE=:CD-ED = l foot, CB=r = 2 feet, EB == V(2^-12) 
= V3, ^5 = 2 VS. Area of triangle ACB = 2VSx J 
=.V3 = 1.732+ feet, (1). By trigonometry, BC : 
sin 90^ : : CE: sin ^^O, or 2 : 1 ; : 1 : .5. But .5 is 
the sine of 30°. Hence the angle BCE = 60^, and 
ACB = 120°, or i of the circle = -jw = area of the 
sector ACBD. Then f7r-\/3[see (1)] = 2.4567 
= area of segment ABD, which is nearly -J- of the 
wheel. 




346. Let PAB represent a portion of the surface of the earth, P being 
the north pole, and AB the equator. Let AF = the distance sailed by 

the ship = 1000 miles, and AE = the difference 
p of latitude. By "plain sailing," we may con- 

sider AFE as a plane triangle, and we have 
B : distance : : cosine course : difference of 
latitude, or 1 : 1000 : : cos 45° : difference of 
latitude = 500 V2 = 707.106 + nautical miles 
= 11° 47' 6", the ship's north latitude. 

347. The equation of the path of a projectile 




J) is y =: X tan a 



gx' 



(1). Transposing, 



2 v'^ cos^ a 
clearing, etc., 2 v'^ cos'^ a{x' tan a — y') = {gx'Y, 



(2) 



Time = t = 



and t'^ = 



V cos a v^ cos^ a 

(see p. 113, Olmsted). Substituting this value 

of t'^ in formula (2'), p. 112, [ tan a = ^, ) we have tan a = — ^ ^^ , , 

y 2r J 2v'^ cos- a 



whence 2 v'^ cos^ a 



gr 



tan a 



Substituting in (2), we have 



gr 



tan a 



(x' tan a — y') = (gx'y 



.r\2 



Reducing, ^ = tan a —~ tan a. Substituting values, we have 



10 

1700 



tan a 



1700 
3500 



tana, whence tan a = .011437 +, which corresponds to 



39' 19'^ = angle. Velocity may easily be found to be 2218.3 feet. 



348. Let ABCD be the barn ; and AG = 100 feet, length of rope. The 
horse is tied at A, and grazes over | of a circle whose radius is 100, or 



MISCELLANEO US SOL UTIONS. 



265 



23,562 square feet, (1). By trigonometry, DF : sin DEF :: DE : sin 

DEE, whence the angle DFE is found to 

be 130 37' 55". Now, 75 cos DEE z= FE = 

72.88 feet, and i>J5' x 72.88 = the area of 

BDF. The triangle BDF — J of the barn — 

975.85 feet = area of DFBC, (2). The angle 



13° 37' 55" = 31^ 22' 



and 



CDF = 450 

900 _ 31° 22' 5" = 58^^ 37' 55" = angle FDH. 

Then the sector FDH, whose radius is DH 

= 75 feet, has an area of 2878 square feet. 

The area of the sector FBG is the same : 

hence the area of both is 5756, (3). Adding the three areas, (1), (2), 

and (3), we have 30,293.85 square feet, the area grazed. 




349. Let FBA be a vertical section through center of glass, C the cen- 
ter of sphere, and CG a radius drawn to BA 
at point of tangency. We have AII= 6, HB 
= 2h,EG=: CD = 2. ^J5 = v6^ + 2.5^z=6.5. 
By trigonometry, AB : sin H:: HB : sin HAB, 
or 6.5 : 1 : : 2.5 : .384615. Hence the angle 
HAB = 220 37' 11". In right triangle AGO, 
sin 22° 37' 11" : 2 : : sin 90° : A C, whence A 
= 5.2. AH-AC=:Q-b.2=.S = Ha Then 
HC + CD=.S-{-2 = 2.S = height of segment 
immersed. In right triangle CHI, IH = 
V2^ - .8-^ = 1.833. 1.8332 x 3. 1416 =: 10.557 + 
= area of base of segment. 10.557 x J of 2.8 
= 14.779, (1). (2.8)3 X .5236 = 11.493, (2). 
Adding (1) and (2), we have 26.272 cubic 
inches. 

Note. — ^C may be found by geometry eb follows: The triangles BHA and AGC 
are right-angled, and have the angle in common. Hence they are similar. Then 
BE: CG::AB:ACy or 2.5:2 ::Q.b:AC, or 5.2. 




350. Let (7 be the center of the 10-acre field, A the stake, and AEHD the 
segment grazed. Put CA = B, AD — 7% and the angle BDA = a. Then the 
angle DBA = a, and the angle Z> CM = 2 a. The triangle BDA is right-an- 
gled, hence sin 90 : BA : : sin a : AD, whence r=2Bsina, (1) . The area of 
the triangle D CE= J B^ sin 4 a, (2) . 4 a : 360 : : sector CDAE : irR^, whence 
4 irB^a 7rB% 
360 



sector CDAE 



90 



(3). Therefore the segment DAEB = 



90 



7?2 



— i it^ sin 4 a = — (ira — 45 sin 4 a) , (4) . In like manner, 180 — 2a 



266 



TABLE BOOK AND TEST PROBLEMS. 



: 360 : : sector DAEH : Trr^, whence sector DAEH = 7rr2(180 -2 a) ^ 

2 /on ^ ^^^ 

Trr (JU - a; ^ ^^y r^^^ ^^^^ ^^ triangle EDA = J r^ sin (180 -2a), (6). 

Therefore segment DHE = ^^'^(^Q - Q^) _ i ,v2 gin (i80 - 2 a), which 

(remembering that the sine of an 
angle is equal to the sign of its sup- 
plement) 

= Jli (90 TT - Tra - 90 sin 2 a) , (7) . 
180 ^ ^ ^ 

By condition, these two segments 

(4) and (7) are together equal to 1 

45 sin 4 a) 

90 sin 2 a) = 160 




acre. 
r2 



7?2 

Therefore — (ira 
90 ^ 



+ ^— (90 7r - Tra 
180^ 

rods. Clearing, we have 2 B^(ja — 45 sin 4 a) + 90 vrr^ — irr^a — 90 r^ 

sin2 a = 28,800, (8). From (1), r = 2B sin a. Then r^ = 4: B'^ sin2 a. 

Substituting this value of r^ in (8), we have 2 B^wa — 90 R- sin 4t a -{- 

360 B^TT sin% - 4 ttZ^^^ sin2a - 360 B^siii^a sin 2 a = 28,800. Dividing by 

2 7?2^ we have ira — 45 sin 4 a + 180 sin^ « — 2 7ra sin^ a — 180 sin^ a sin 2 a 

14 400 

r= — ? = 9 TT, (9). Erom trigonometry we have sin 2 a =: 2 sin a cos a. 

Hence 45 sin 4 a = 90 sin 2 a cos 2 a. Using this, (9) becomes ira — 90 
sin 2 a cos 2 a + 180 tt sin^ a — 2 7ra sin^ a — 180 sin"^ a sin 2 a = 9 tt, (10). 
From trigonometry we have sin^ a = J(l — cos 2 a). Substituting in (10), 
we have ira — 90 sin 2 a cos 2a + 90 tt — 90 tt cos 2a — ira -\- ira cos 2 a 
— 90 sin 2 a + 90 sin 2 a cos 2 a = 9 tt. Collecting or cancehng terms, we 
have — 90 sin 2 a — 90 tt cos 2 a + Tra cos 2 a = — 81 tt. Changing signs, 
90 sin 2 a + 90 TT cos 2a — ira cos 2 a = 81 tt. Dividing by tt, 28.65 sin 
2a + 90 cos 2 a — a cos 2 a = 81. Factoring, 28.65 sin 2 a -f cos 2 a (90 — a) 
= 81. Solving this equation (see Ray's "New Higher Algebra," Art. 
436), we find 2 a = 27^^ 18'. Then a = 13° 39'. Then in triangle BDA, 

sina:r::l:2/?, or sin 13° 39' : r :: 1 : 2^/^552 = _?5_ 

' TT 

rods, the required length of line. 



V7 



whence r = 10.6 + 



351. Let A and D be centers of the circles, BC Si perpendicular to AE. 
Since DE is about f of a rod, BC more than 7 rods, and AE nearly 32 
rods, it is evident that the angle BDE is obtuse. Put ^J5 = 31.91538 
= B, DB=: r, the angle BAD = 6, DE = .60606 of a rod = a. Now, 
AC= Bcose, Siiid BC= BsinS. Hence CD=AE-AC- ED =B -B 
cos d — a. Again, should we join the middle point of the arc BHE 
with the center A, we should have half the chord BE as B sin J 6. 



MISCELLANEO US SOL UTIONS, 



267 



Hence BE = 2 R sin} d. By Proposition 13, Book IV., Loomis's 
" Geometry," BE'^ = BD^ + DE'^ + 2 DE 
X CD ; that is, 4 m sin^ J ^ = 7-^ + a^ + 
2a{R — Rcosd — a), (1). From trigo- 
nometry we have 2 sin"^ } 6 = 1 — cos ^. 
Putting this equivalent in (1), we have 
4i22sin2i(9=rr2+aH2a(2i2sin2J^-a), 
(2). Developing (2), etc., we get 

^2 = 4/^2 sin2 }e - 4 aZ2 sin2 J ^+ a^, 
or r2 r= 4 /?(Z^ - rt) sin2 J (9 + a^. Therefore 
r = V{4 i^(/^ - a)sin2 i ^ _^ a% (3). Area of sector ABHE = ^, (4). 
Area of triangle ABE = ^ AE x BC = "^^ , (5). Then area of seg- 
ment BHE = (4) - (5) = - ^^ C^ ~ ^^^ ^) , (6). Area of triangle BDE = 




BE X BC ^ aR sm (9 ^ ^-.^^^ ^^ trigonometry the sine of the angle BDA 

A A 

- BC -r- BD ^ ^lI}^, (8). Taking the inverse functions of each mem- 

r 
ber of (8), or, in other words, measuring the angle BDA by its subtend- 
ing arc, we have, angle BDA = sin-^ / — ^Hl_ j . But this is the measur- 
ing arc of the angle BDA when radius is unity. Hence, when radius is 
DO = r, the measuring arc of the angle BDA is arc OKB=rsiii~^ ' 



Now, by the same principle with which equation (4) was obtained, we find 

the area of the sector DO KB — 7' x Jrsin""^ ( ^ — ] = J ?'2 sin~i ( 

(9). Evidently, KB HE = half the area grazed, which is correctly ex- 
pressed by the sum of (6), (7), and (9). Therefore ^^(^ - sm 6) 
+ ^^ f^ ^ + 1 r2 sin-i/'^^JHLi'j = 80, (10); and R%d - sin^) + aBsin 

+ ?^2 sin-i[ ^ ^^^^ \ = 160, (11). Substituting in (11) the value of r 

from (3), we have R^(d - sin 6) + aR sin 6> + {4 R(R - a) sin2 ^ e -^ a^ 
( Rsind 



X sin-i 



= 160, (12). Solving (12) by 



( Vl^R(R - a)sin2i^ + a^] 
*' Position" (which may require several hours), we find 6 = 18^ 15' 9.4'^ 
Substituting in (3), we find r = 10.04588 rods. (From B. F. Burleson.) 



268 TABLE BOOK AND TEST PROBLEMS, 



SOLUTIONS INVOLVING CALCULUS. 

1 
352. Let X = the number. Then x^ — x = a maximum = y. Differen- 
tiating, ( -x'' — 1 ]dx = dij. Therefore -^ = -x"' —1 = 0, and x'^~ = n, 

whence x = n^-"', Ans. When n = 2, we have x = 2^-2 = 2'^ = — = -. 

3 2'^ 4 

When n = S, we have x = 3^^ = 3"' =~= ~^- = -4=- When n = ^, 



3! V3^^ V27 



4 

♦ — — -"^11 1 

we have x = 4^-'* = 4 ^ = — = — — =: — -— -. When n = 5, we have 

4I ^^4^ ■v'256 

— -^ 1 1 

X = 51-^ = 5 ^ = — = , etc. 




5 1 \/3125 

353. When the hypothenuse is a minimum, the base = altitude. Hence, 
in the figure, AC = CB, and the angle B = angle A = 45°. Bisect the 

right angle by CD. Then the angle ACD = the an- 
gle A, and CD = DA, and AE = EC. But EC = 12. 
Hence AC =24:, and OjB = 24. This result is veri- 
fied by the differential calculus, as follows: Put J.C 
= x, BC=y, EC =12 = a. Then AE = x - a. 
By similar triangles, we have the proportion AE: ED 
: : AC: BC, or x^ a: a:: x: y. Hence xy — ay = 
ax, (1). By the problem, x' -f if- = AB^ = a mini- 
mum, (2). Differentiating (1) and (2), we have xdy -\- ydx — ady = 

adx, (3), and 2 xdx + 2 ?/cZi/ = 0, (4). Erom (4), dy = - — • Substitut- 

'y 

X d IT nxdx 
ing this value of dy in (3), we obtain \- ydx -\ = adx, whence 

y y 

y'^ — ay ^x'^ — ax. Completing squares, y'^ — ay -{-\cfi ^x'^ — ax + \ a^. 
Extracting square root, y — ^ a = x — ^ a, whence y = x. That is, when 
the hypothenuse is a minimum, the base = the altitude, or perpendicular. 
Hence the construction and solution are same as above. 

354. Let X = the height, r = 2\/2 = radius. Then x"^ + ?*- = the square 
of the distance. Therefore the light varies as By trigonometry, 

X2 4- ?'2 

we have Vx"^ + r'^ : 1 : : x : sin ^, d being the angle of inclination to the 
plane, whence sin d = — ^ , which also expresses the variation of 



the light. Therefore the light varies as 



Vx^-f 

varies as 

X X 

= y' 



x2 + r2 Vx'^ + r^ (x'^ + r^)^ 



MISCELLANEOUS SOLUTIONS. 269 

Differentiating, g^ ^ (x^ + r^) ^ - x\l (x^+_r^)J2x} ^ q^ (.^^^^. ^^^ 

dividing by (x'^ + r^Y, we have 2 x"^ = r^^ whence x = ±-^. By testing, 

we find that x — ± — = corresponds to a maximum. In the special 

/- ^"2 
2v2 
problem x = ± — 3- = di 2 rods, the required distance. 

V2 



PROMISCUOUS SOLUTIONS. 

355. Let X = .45, the repetend. Then 9 + x = the given number. 
100x = 45.45, and 100 x - x =:= 99 x =: 45.45 - .45 = 45. Then x = || 

= -j5_ ; and 9 + X = 9 + y\ == 9i\. 

356. 1100 X 8 = 8800. 8800 - 5280 =r 1.6+ miles. 
Note. — Sound travels through air about 1100 feet per second. 

357. Every time the wheel revolves, the nail-head describes a cycloid, 
whose length is found by dividing four times the distance by tt. Hence 
3.1416 X 4 - 3.1416 = 4 rods. 

358. First Solution. Suppose A works x times as fast as B. Then 
digging must be x times as difficult as weeding. By second condition, 
the faster man has the easier work, and finishes four times as many 
rows. Hence x x x = x^ = 4, and x = 2. If A works twice as fast as 
B, he should receive $4, and B -'^2, per day. 

Second Solution. Let A's rate be x times B's. Then A's rate : B's 
rate : : x : 1. By second condition, A's rate : B's rate : : 4 : x. Hence 
X : 1 : : 4 : X, whence x^ = 4, x = 2, and A should receive | of $6, or ^4. 

359. Let X = number of pounds of gold. Then a — x = number of 
pounds of silver. 1 pound of gold loses - pounds in water. Hence x 

HOC - w 

pounds lose — pounds. 1 pound of silver loses , and (a — x) pounds 

f Q^ X^ P 

lose -^^ ^ pounds. Since the whole loss in water is m pounds, 

nx (a — x)» , a(m — p) ., a(n — m) 

. ^ ^ £_ — ^yi whence x = — ^—, and a — x = -- 

a a n — p n — p 

360. First Solution. In 1 day A and B dig y\^ b and C ^^q, and A and 
C J5. Adding, 2 (A and B and C) dig | in a day, or all dig J of J = Jq in 
1 day. Hence i^ -4- J^ = 10 days, the time in which all can dig it. Then 
1 -^ (yV — T2) = ^0 days = C's time, 1 -f- (^V — J^) = 20 days = A's time, 
and 1 ^ (j\ - J3) = 30 days = B's time. 



270 TABLE BOOK AND TEST PROBLEMS. 

Second Solution. A and B dig -^^ in 1 day, and A and C ^^. Hence 
B digs -^^ — J- = g^o more than C. But B and C dig g%. Hence C digs 
J(A - A) == e'o in 1 day, or the ditch in 60 days ; B digs ^V - A = io 
in 1 day, or the ditch in 30 days ; A digs ^2 ~ to = 2V ^^ ^ ^^y> ^^ ^^^ 
ditch in 20 days. 

361. Let A, B, and C be the lower ends, and the point beneath the 
tops. 0, being the center of the triangle, is f of AD., a perpendicular. 
A But AD - VAB^ - BD^ = V2700 = 61.9 + 

feet. Hence ^0 = t of 51.9 = 34.6+ feet. 
This is the base, and the pole erected at A 
is the hypothenuse of a right-angled tri- 
angle, of which the required plumb-line is 
the perpendicular. Hence the required line 
= V(50^-34.6^)rr: 36.1 feet, nearly. 

362. The semi-annual interest on the 
bonds is 3i per cent of $7000 = -$245, to 
secure which at 8 per cent would require $245 -i- .04 = .$6125. This 
would allow me the interest on the bonds and $6125 of their face value 
at the end of the 20 years. To be entitled to the remainder, I must 
invest an amount equal to the present value of $7000 — $6125 = $875 
for 40 intervals at 4 per cent compound interest, or $875 ~- 4.8010206 + 
= $182.25. I therefore paid $6125 + $182.25 = $6307.25. 
Note. — The 4.8010206+ is the 40th power of 1.04, or 1.044o. 




Let ic = ?/ + 3. Then x^ = 2/2 + 6 ?/ + 9, and Qx=i^y + 18. Sub- 
stituting, the equation becomes ^/^ + 6 ?/ + 9 — 6 ?/ — 18 = 7, or ?/2 = 16, 

?/ = 4, and x = 4 + 3 = 7. 

364. Let X = ?/ - 1. Then x^ = ?/2 _ 2 ?/ + 1, and 2x = 2y -2. Sub- 
stituting, we find 2/'^ = 9, ?/ = 3, x — 2. 

365. A square field containing 10 acres is VlOO x 10 = 40 rods square, 
and the fence is 160 rods in length. In the rectangular field the length is 
4 times the width. Hence the width multiplied by 4 times the width = 4 
times the square of the width = the area = 1600 square rods. Then the 
square of the width = 400 square rods, and V400 = 20 rods = the width ; 
and the length = 20 X 4 =: 80. The fence is 2(80 + 20)= 200 rods 
long ; and 200 — 160 = 40 rods, the difference. 

366. First Solution. Assume x^+x = a-x^. Thenx = — Assum- 

d^— 1 
ing values for a, we easily find x ; and substituting these values of x in 

a;2 + X, we obtain square numbers. 



MISCELLANEOUS SOLUTIONS. 271 

Second Solution, x"^ + x = a square number. Assume x^ -\- x = ni^. 
Then xr- — ni^ — x. Put m^ — x = any square number, less than 9?i-, in 
terms of m and x. For instance, (in — x)"^ < m^. Then x"2 =: m^ — 2 mx 
-\- x'^, x = m — X, or m = 2 ic, m^ = 4 x'^. Or x-^ + x = 4 x'^, x -{- 1 = 4 x, 
or 3 X = 1, X = J. Vakies of x may be found at pleasure. 

(From Dr. I. J. Wireback.) 

367. Let m and n represent the stumps and stones that give an earning 
of 38 ; and AI and N, those that give $12. Then 2bm + in = 800, or 
50m -\-n = 1600, (1) ; and 25 ilif + J iV ::= 1200, or bOM -{- N=: 2400, (A). 
Since 1 stump requires the time of 40 stones, the relative times may be 
represented by 40 m + n, and 40 M + N, so that 40 m + ?z : 40 M + N 
: : 3 : 3| : : 4 : 5 ; whence 200 m + bn = 160 M^4 N, (2). Subtracting (2) 
from 5 times (1), we get 160ilf + 4iV^ + 50 m = 8000, (3). Subtracting 
(3) from 4 times (A), we get 4 3/= 5m + 160, (4). We have four 
unknown quantities, and only three equations : hence the problem is 
indeterminate. From (4), M= J(5?9z+ 160), (5), an integer; and ^ m 
must be an integer, say p. Then m = 4tp. The lowest value assignable 
to p is 0, in which case m — 0. Substituting in (5), M — 40, and in (1), 
gives n — 1600 ; also M in (J.), gives N — 400. The next higher value 
of p is 1, making m = 4, n= 1400, 3/ =45, and N— 150. The next 
value of m is 8, which makes N — — 100, which is not admissible. As 
all succeeding values of N are negative, there can be but two combina- 
tions, making stumps and 1600 stones to earn $8, and 40 stumps and 
400 stones to earn %V1 ; or 4 stumps and 1400 stones to earn $8, and 45 
stumps and 150 stones to earn %Vl, Consequently 40 stumps and 2000 
stones, or 49 stumps and 1550 stones, will satisfy all conditions. 

(From William Wiley, Detroit, Mich., in the School Visitor.) 



272 TABLE BOOK AND TEST PROBLEMS. 



^.r H' 






ai'^l . 9r,^ ^ CURIOUS RESULTS. 
'^- DIGITS. 



368. To get any required digit in the product, multiply the given num- 
""Tber by 9 taken a number of times denoted by the digit required. Thus, 
to get all 4's, multiply by 9 x 4 = 36. 12,345,679 x 36 == 444,444,444. 



OfiQ ^ —. f 2 \Z . 512 _/8N3. 8 _/^2\3. onH 

OD». 2¥T3 75 6 9 — \.2J2) ? 13^9 76 " ^76^ J 3'2 4 6T75"9 — K^T^J J *^^^ 

JL2_5 _ 

¥3 8 9 76 



i2^_ - (^%y 



"ONE CENT." 

370. We will make the calculation by means of logarithms, and hence 
will give results in round numbers. The amount of $1 for 1 year at 
6 per cent is $1.06; for 2 years it is ($1.06)2; for 3 years, (.$1.06)3; 
and so on. Hence the compound amount of 1 cent for 1881 years is 
^ = .01 X (1.06)1881. Applying logarithms, we have log A = log .01 + 
1881 log 1.06 = - 2 + 1881 X .02530 - 586526 = 45.600333. Therefore 
A = $3,984,130,000,000,000,000,000,000,000,000,000,000,000,000,000, which 
may truly be called a "round " sum. 

(From the late E. B. Seitz, in Mathematical Magazine.) 

INVOLUTION OF IMAGINARY QUANTITIES. 

371. Squaring, — a'^ = x'^. Squaring again, a^ = x^. Extracting fourth 
root, a = X. But Va' = a, and \/— a'^ = x. Therefore Va- = V— a^, 
or a^ = — a'^, or 1 = — 1. The beginner will understand this better by 
studying the involution of imaginary quantities. By definition, the 
square root of any quantity multiplied by itself should be the quantity 
itself. Hence V— a x V— a = — a. But if we multiply the quantities 
under the radicals, we have V— ax V— a = Va^ = ± a ; that is, there 
appear to be two products of V— a x V— a. The true value of Va'^ 
in this case, however, is — a. We know this, because the factors that 
produced a^ are known. Were the factors not known, the value of Vo^ 
would be in general ± a. 

372. Since squaring a riidical removes the symbol V, we have 

(Viri)2^ - 1, (1). (v-^)^= V-1 X V^i xV^^i =:(\/^)"-2 X 
V- 1 := - 1 V^:^, (2). qV- ly = ( V- i)nV- 1)-^ = (- 1) X (- 1) 
= +1, 03). 



CURIOUS BESULTS. 273 

373. It is because the fewer times a negative quantity is taken, the 
greater will be the product. This will be clearly seen from the following : 

-7x 3 = -21, 

^ _ 7 X 2 =: - 14, 

-7 X 1 =- 7, 

- 7 X =: 0, 

- 7 X - 1 = 7, 

- 7 X - 2 = 14, 

_ 7 X - 3 = 21, etc. 
We notice that each product is 7 greater than the preceding, and that as 
the multiplier decreases, the product increases. Or take the expression 

— b X (a — c) . This may be written as follows : ( — 6 x a) — ( — & x c) , 
which is equivalent to — ab —(— be). But minus a minus quantity = 
phis the quantity. Hence — ab — {— be) — — ab + bc^ from which we 
see that ~ b x — c = -\- be. 

THE ZERO FACTOR. 

374. First Solution. If a = x, then a^ = cc'^, and ax = x'^. Subtract- 
ing x^ from both sides, we have ax — x'^ = xr- — x- = a- — x'^ (since a^ — x^). 
Factoring, x{a — x) = {a -\- x) (a — x), (1). Dividing by a — x, we get 
x = a-{-x = '2x. Whence 1 = 2. 

Note. — While all the operations appear to be legitimate, the result is evidently 
absurd. The error must be the using of the zero factor (a — x) in dividing (1). 
Instead of striking out the common factor in (1), we may indicate the division thus: 

— ^ = -^ ^ IN o w, since a = x, a — x = 0, and the fractions become n = §, 

a—x a—x 

each member consisting of the symbol of indetermination. 

Second Solution. If a = x, then a^ = x"^. Transposing, a^ — x'^= 0, (1). 
Factoring (1), (a + x)(a — x) = 0. Hence a + x = 0, (2); and a — x 
= 0, (3). From (2) we see that 2 a, or 2 x, = 0, (4). If a = x = 1, 
then (4) becomes 2 = 0. Ifa = x = 2, then (4) becomes 4 = 0, etc. 

375. This proportion must be true, since the product of the extremes 
is equal to the product of the means. Now, as a is greater than 0, it is 
therefore greater than — a. But if in the first ratio the second term is 
the greater, it must also be the greater in the second ratio. Hence we 
have these two inequalities: a> — «, and — a > a. If a = 1, then 
1 > - 1, and - 1 > 1. If a = 2, then 2 > - 2, and - 2 > 2, etc. Again : 
solving the proportion, we have (— a)'^ = a'^. Dividing by — «, we get 

— a = -^, which may be written — Now, the quotient of 

— a a — 2a 

«2 -f- (a — 2 a) is a + 2 a + 4 a + 8 a + 16 a + 32 a + 64 a, etc. ; that is, 

— a is infinitely greater than -\- a. 

ELLWOOn's TEST PROB. 18. 



274 



TABLE BOOK AND TEST PROBLEMS. 



SOMETHING TO INVESTIGATE. 

376. 127,364 x 3 i= 382,092. 3 + 8 + 2 + + 9 + 2 = 24. 2 + 4 = 6. 
Starting with this, we may proceed as follows, grouping the nine digits 
thus: 1, 2, 3; 4, 5, 6 ; 7, 8, 9. Take any number and multiply by the 
first digit of each group, proceeding as above, then by the second and 
third, as follows : — 

8425 8425 8425 

14 7 



8425 = 19 = 10 = 1 


33700 = 13 = 4 


58975 = 34 = 7 


8425 

2 


8425 
5 


8425 
8 


16850 = 20 = 2 


42125 = 14 = 5 


67400 = 17 = 8 


8425 
3 


8425 
6 


8425 
9 



25275 = 21 =3 



50550 = 15 = 6 



75825 = 27 = 9 



In this case the resultant digits are the same as the multipliers, and 
the nine digits occur in regular order when taken vertically. Likewise 
when the multiplicand is 28,747, etc. Taking 25,432, we have the 
f ollowinof : — 



25432 
1 



25432 
4 



25432 

7 



25432 = 16 = 7 


101728 = 19 = 10 = 


1 


178024 = 22 = 4 


25432 

2 


25432 

5 




25432 

8 


50864 = 23 = 5 


127160 = 17 = 8 


203456 = 20 = 2 


25432 
3 


25432 
6 




25432 
9 



76296 = 30 = 3 



152592 = 24 = 6 



228888 = 36 = 9 



The resultant figures are the same as in the previous examples, and in 



the same horizontal line, though not in the same order 



THE PROPOSITION OF ARCHIMEDES. 

377. ' ' This he might easily have done could he have brought his lever 
to bear upon it ; for it rests upon nothing, impinges against nothing, and 
floats in space, a body perfectly free to move in any direction. His lever, 
therefore, would have been a useless thing, as the slightest force brought 
to bear upon it would have caused it to move. He need only have 
stamped his foot, and the ponderous globe would have moved obedient to 
the impulse. His idea of the subject must have been that the world 
rested in all its mass like a rock upon some other ponderous body, and 
that he could apply a sufficient force by his leverage to lift it up and over- 



CURIOUS RESULTS. 



275 



turn it. His calculations and conclusions were undoubtedly correct, but 
the element of time he overlooked in his computations. Calling the 
diameter of the earth 7920 miles, and each cubic foot of its volume 
to weigh, as has been estimated, 300 pounds, we find that the earth would 
weigh 5,765,171,439,574,305,792,000 tons. Supposing Archimedes could 
exert a continual force of 30 pounds at the end of his lever, we find that 
one arm of the lever must be 384,344,762,638,287,052,800,000 times longer 
than the other in order that he might move it. Hence in order that he 
might move the earth to the height of one inch, it would have been neces- 
sary for him to have moved with the long arm of his lever 384,344,762,- 
638,287,052,800,000 inches. Kow, constantly pulling with a force of 30 
pounds, he could not, with his single- man power, have traveled more 
than 10,000 feet per hour ; and at that rate, too, not more than 10 hours 
per day. He could therefore, at his utmost, have moved his end of the 
lever but 100,000 feet per day. Hence it may be readily calculated that 
to have raised the earth only one inch, it would have required his con- 
tinual labor for 8,774,994,580,737 centuries." Admitting this, we must 
deny his ability to ' ' move the world, ' ' not on account of a lack of power 
or principle, but on account of the shortness of human life. 

1888. 
378. 



487 


459 


461 


481 


465 


477 


475 


471 


473 


469 


467 


479 


463 


483 


485 


457 



SUMMATION BY SUBTRACTION. 

379. Assume any number greater than the sum, and from this subtract 
one of the given numbers ; from the remainder take another of the 
numbers ; and so on till all have been subtracted. Then take the last 
remainder from the assumed number. 



Assume 



10000 
379 

9621 

8452 

1169 



1169 
31 

1138 
60 

1078 



10000 

107 8 

8922 Ans. 



276 TABLE BOOK AND TEST PROBLEMS, 



SOLUTIONS TO SERIES. 

380. FirsL sum the series — = \- — = 1 = 1 

1.2.32.3.43.4.5 

Let — = the nth. term of the series = , and Sn = the 

f^n n02+l)(n + 2)' 

sum of n terms of the same. 
We shall have & == — + c, 

2fXn 

and Sn+i = ^--:^+c = -^ ?^ti^ + c 



Therefore c 



2/Xn+i 
3 



and Sn 



2fxi 2^n 2(1.2.3) 2n{n-^\){n + 2) 

= [when 72 = 00]-. 

4 2(?2 + l)(n + 2) *- -^4 



Seco7id, sum the series 



1 +^^^+^j^+ 



1.2.3 3.4.5 5.6.7 
In the series for loge (1 + x), put x = 1^ and we have 

2 3 4 5 6 1.2 3.4 5-6 

1 1 1 



= 1- 

2.3 4.5 6-7 

Therefore 2 loge 2 = 1 + — - — + — - — + — - — + - 
"^ 1.2.33.4.55.6.7 

Whence — ^— + — —- + — — + ... = loge 2 - -• 

1.2.33.4.55.0.7 2 

Einally, by subtracting this second series and its summation from that 
of the first, we obtain 



1 +_4_, + ^,X^+.._3_log„2 



2.3.4 4.5.6 6.7.8 4 

= .05685281944005469058276787876403 + 



SOLUTIOJSfS TO SERIES. 277 

(correct to 32 decimal places, as we have computed the Napierian loga- 
rithm of 2 to that number of decimal places) . 

KoTE. — Logg = "CTapierian logarithm. 

381. Put 

7^ + (y^ + l) + (n + 2)4-(?^ + 3) ^^ B C D .^. 

n(n + 1) (n -f- 2) (?i + 3) n n + l ?z + 2 ?i + 3 ^ ^ 

Clearing (1) of fractions, 

4 71 + 6 = ( J. + 5 + C + D) ^3 + (6 ^ + 5 5 + 4 a + 3 2)) ,22 

+ (11^ +6^ + 3C+2Z))7i + 6^ (2) 

Equating coefficients of like powers of n in (2), and solving the equa- 
tions, we find that 

A=l, B = ~\, C = -\, and D^ 1. 
Therefore 

ri + (72 + 1) + jn + 2) + Qi + 3) ^ 1 1 1, 1 . 

7i(?z+ l)(?i + 2)(?^ + 3) 71 7i+l 71 + 2 ?2 + 3' 

that is, the sum of the given series is equal to the sum of the two series 

whose general terms are - and , minus the sum of the two 

n 71 + 3 

series whose general terms are — and 



71 + 1 72 + 2 

Therefore 



„.1 + _L\ 

71 71 + ly 



uu...i+^+ 1 



71 7i + 1 72 + 2 



4 72 72+1 71 + 2 72 + 3 

1-1-^- + -^ = ? ? ^Ans, 

3 72+172 + 3 3 (7l+l)(72 + 3) 

When 71 — 1 , we have 8i — y%. 
When 72 = 2, we have S2 = td- 
When 72 = 10, we have Sio = f|f . 
When 72 = 1000, we have Sim = f §f IHt- 
When 72 = 00, we have S^ = f . 



278 TABLE BOOK AND TEST PBOBLEMS, 

382. Let X and y = the extremes of the proportion, and z and w = the 
means, or vice versa. We shall then have 

xy = zw (1) 

x-\-y -{- z -\- w = a (2) 

x^ + if-{-z'^-\-w''- = b (3) 

x^ -^ y^ -i- z^ -\- w^ = c (4) 

From (2), z -\- iv = a - (x -\- ij) (5) 

(5)2 = z'^-j-2ziv-{-w'^ = a'^-2 a<ix + ?/) + x^ + 2 x^/ + 2/^ (6) 

Suppressing 2 ^'i^ in the first member of (6) and its equal 2xy m the 

second, and then adding x^ + y^ to both members, we have 

x^ + if + z'^ -j- iv'^ = h = a^ - 2 ai^x + y)-^ 2x'^ -i- 2 if (7) 

From (7), 2 a(x + y)=a^-b-^ 2(^2 + i/) (8) 

(5)3 =: ^3 + 3 ^wj(^ + i(?) + lo'^ = a3 - 3 a2(x + ?/) + 3 a(x + 2/)2 

-x3_ 3^2/(^ + 2/)- 2/' (9) 

Transposing terms in (9), and observing that Sziv = Sxy, we have 

x^-^y^-{-z^-\-w^-\-Sxy(x-\-y^z + w) = a^-Sa%x-]-y) + Sa(x+yy; 
that is, c + Saxy = a3-3a2(x+?/)+3a(x'H2/2)+6ax?/, 

or c-Saxy = a^-S a^(x + ?/) + 3 a(x2 + y'^) (10) 

(8)x^ = 3a2(x + 2/) = 5|-'-^ + 3a(x2-f 2/2) (11) 

(11) - (10) = ^^ ~ ^ ^^ =: 3 axy -c (12) 

From (12), x?/ or ^ii? = Q^^ - 3 ^^ + 2 c ,^3. 

6a 

4(13) + (8) = 2 a{x + 2/) + 4(a^ - 3«?> + 2 c) 

6a 

= a2- 5 + 2(x + ?/)2 (14) 

Resolving (14) as a quadratic, we obtain 

x^y and ^ + i/;:r.i a ± lj/8c-^^a?> + a^\ ^^^>^ ^^^ ^^g>^ 

From (13) and (15), and from (13) and (16) we easily find x and y, and 
also z and w 

-1^.1 //8c-6a& + a3\ , 1 /r9a6-2a3-4c ^i^ //8c-6a?> + a3\-i 
-^^^Wl Va j=^2VL ^a ^^NV ^a )\' 

Axis. 

The sign before the first and third radical must be taken as positive for 
determining x and ?/, and as negative for determining z and i(?, or vice 



SOLUTIONS TO SEBIES. 279 

versa. If we take a = 21, h — 125, and c = 819, we shall find by the 
formula that x = 8, y = ^-, z = 6^ and to = 4 ; and the proportion is 
8 : 6 : : 4 : 3. 

383. We have 

a-\- (a+d) -\- (a + 2d) + '" +[a+(n-l)(^]=Jw[2a+ (H-l)d]=s (1) 

= [by expanding the terms, and summing those that are similar] 
na^ + n(n -l)ad-{- l n{n - l)(2n - \)d^ = b (2) 

a^ + (a + dy+ (a + 2 dy + ... + [a + 0^ - l)dY 

= [by expanding the terms, and summing those that are similar] 
na^-^l7i(n-l)a''^d+in(n-l)(2n-l)ad^-\-ln^(n-iyd^=c (3) 
Combining (1) and (2), we find that 

^^ 2 USnb-3s^\ (4) 

and ^^a^l /r(3n6-3s^)(n-l)-| ^g^ 

Substituting in (3) the values found for d and a, we obtain by develop- 
ing the quadratic equation, 

en"- -Ssbn=-2s^ (6) 

Therefore, by resolution, 71 = ^[Sb — ^/(db'^ — 8sc)] = [by substitut- 

2 c 
ing the numerical values of the letters] 12. By substituting the numer- 
ical value found for n in (4) and (5), we find that d = ±11, and a = 35 
and 156, the first and last terms of the series. By taking a = 35, and 
d = 11, the series will be ascending ; but if we take a = 156, and d = — ll, 
the series will be descendino-. 



384. We have 



a-\- ar-i- ar"^ + ••• ar^-^ = ^-^^ ^ = s (1) 

r — 1 



-1 — <^(^^ — 1) 
~ r~l 

«-2 _^ ^2^2 + ^2^i + . . . ^2^2n-2 ^ a^fr^n - 1) ^ ^ ^g) 



r^ - 1 



a3 + ^3^.3 4. ^3^.3 ^_ ... ^3^n-3 ^ ^^!(r^! 11 ^ c (3) 

r^ — 1 

(2)^(1)= a{rn+V) ^h (4) 

r+ 1 s 



280 TABLE BOOK AND TEST PBOBLEMS, 

Combining (1) and (4), we find tliat 

2s ^ ^ 

and r« = ^'^^^' ~ ^^ (6) 

^ ^ ^ ^ r^ + r + 1 s ^ ^ 

Substituting in (7) the values found for a and ?*", we obtain, by develop- 
ing the quadratic equation, 

(s* _ 4 sc + 3 62)r2 _ 2(5^ + 2 sc - 3 &2)^. ^ _ (^4 _ 4 sc + 3 62) (8) 

Therefore, by resolution, 

r = {s* + 2 sc - 3 62 4. ^[12 s(sc - &2) (^3 _ c)]}-^ (s* ^ 4 sc + 3 62) 

= [by substituting the numerical values of the letters] 1 J or f . 

The former value gives an ascending series, and the latter a descend- 
ing one. By substituting the values found for r in (5), we find that 
a = 64: and 729, the first and last terms of the series. We finally have 

_ log[a + (r- l)s] -logg _ 7 . 
logr 

and the series is 64, 96, 144, 216, 324, 486, 729. 

385. The series is 55, 1079, 4599, 12151, etc. 

Mrst order of differences = 1024, 3520, 7552, etc. 

Second order of differences = 2496, 4032, etc. 

Third order of differences = 1536, etc. 

The first terms of the several orders of differences are D^ = 1024, 
J>2 = 2496, Dg = 1536, and the first term of the series = a = 55. Sub- 
stituting these values in the formula 

s = na+ <''-^^ D,^ '<''-^^^''-^^ D,-\- ''^''-^^^''-^^^''-^^ D,+ etc., 
1-2 ^ 1.2.3 ^ 1.2.3.4 ^ ' 

we obtain, by reduction, 5 = 64 ?i4 + 32 n^ — 32 ii^ — 9 n. 

Factoring, s = 16(2 ii^ + J 11)2 - 18(2 if- + J n) (1) 

Solving for s = 22395834549559, 

^ , o •' , 1 18929704 

we find 2 }i^ -\- In = » 

^ 16 

whence n = 769. 



SOLUTIONS TO SEBIES. 281 

To make tlie solution general, consider w in (1) as the unknown quan- 
tity, and resolve as a double quadratic. This gives 



n = iC^lQ + 2 Vl6 s + 81 - 1) 
= [when s = 65, 1134, and 22395834549559 respectively] 1, 2, and 769. 

386. First Solution, (a) Let fXn = the nt\\ term of the series 

Therefore fin — A*n-i = - Kf^n-i - Mn-2), 

whence /Xg — yu^, fi^ — At2, /-t^ — ^tg, etc. , is a geometrical series with the 
common ratio — J. The first term of the series, or /jl,^ — /Xj, = b — a, 
where a = 1 (the first term of the series), and & = 19 (the second). The 
last term of the series, or 

/i^-/A,^_i = (5-a)(- 1)^-2. 

Whence, by addition, 

Mn - Ml = (& - «)[1 + (- J) + (- iY + - + (- J)"-"] 

= [by summing the series in brackets, etc.] ^ ~^^ [1 — ( — J)"~^]. 

Therefore 

= [when a = 1, and 6 = 19] 13 - 12(- i)«-i, Ans. (1) 

(&) Giving to n in the value found for the general term all possible 
integral values from 1 to n^ we obtain two distinct geometrical progres- 
sions : one for the odd powers of — J, and the other for the even powers 
of — J. Summing these two series and adding them, then substituting 
the result in formula (1), after multiplying the last term in it by n^ we 
obtain for the required sum s„ of the given series, by simplification, 

Sn =in(2 h-ha)-i(b-a)ll-(- J)-] 

r=:[when a = l, and b = 19'] VSn - S + 8(- i)^ (2) 

We find by formula (2) , when 

f 71 = 1, Sn=l ] { 71 = 4:, s^ = 44J 

\ n = 2, s,=20\ \ n = 7, s, = 821f 
[n = Z, s, = 30 J L n = 12, s,, = US^{^ 

n = 500, S500 rr 6492 + 1 - 26187124863169134960105517574620793217733 
1363683445183158663309447690703712373964390661607386072332572070934 
73020480568073738052367083144426628220715008, exactly. 



282 TABLE BOOK AND TEST PROBLEMS. 

Second Solution. This is a recurring series of the second order, whose 
scale of relation is m^ n(= J), and is equal to two geometrical recurring 
series of the second order, in each of which the scale of relation is m, n. 
Let X and y be the first terms of these series. We then have 



x-^ xB + xB^ + xB^ + xB^ '" xB 



n-l 



and ij + ijB, + 2jB,^ + yB,^ + yB,^ •.• yB{^-\ 

Adding, we have 

(oc + y) + (xB + 2/i?i) + ••• (xi?"-i + 2/i?i«-i), 
a recurring series of the second order, whose scale of relation is m, n. 
We now have x + ?/ = 1, and xB + yB^ = 19, 

whence x = l?_=l^, and y = ^-=1^5. 

B - B^ i? - i?i 

In a geometrical series that is also a recurring series of the second 
order, as x -}- xr + xr'^ + xr^ + etc. , we have 



xr^ = mx + nxr, whence r = J(?2 ± V-i 77i + n'^). 

In our problem, m — n — J. Hence r = 1 = i?, and — \=z B^. 

Using these values in expressions for x and ?/, we have 

^ = 19^(^11 = 13, and 2, = ^^i^ = - 12. 

The nth. or general term is 

xB^-^ + ?/i?i^-i = 13 i2«-i - 12 JR^^-i = 13 - 12(- J)«-i. 

387. (a) Put a = 4, and 6 = 64, the first two terms of the series 
respectively. Let fXn = the nth term of the series ; then will 

/x„ - V/x,^_j //„_9, and log fin = i(logfMn-i + log/x„_2). 

Hence, with logarithms, the problem becomes exactly similar to the 
preceding one, and we have 

log/.n = loga + |(log5-log«)[l-(-i)"-i]. 

Whence, by reverting again from logarithmic functions to natural ones, 
we have 



fMn= a 

a 



= [when a = 4, and h = 64] 4(16)^i^i-(- ^^'' ^\ Ans. (1) 

(b) Giving to 7i in the value found for the general term all possible 
integral values from 1 to ti, and taking their continued product by 
adding the resulting exponents, etc. , we obtain 



a 



SOLUTIONS TO S^BIES, 283 



&\|[3n-2-(-i)«-l] 



= [when a = 4, and 6 = 64] 4«(16)^t^^^- 2- (-^)^ ^] (2) 

We find, by formula (2), when 



{ n = l, Pi = 4 



[ n = 3, Pg = 4096 J 



f w = 4, P, = 131072 1 



^' n = 2, P^ = 256 m' ^^ = '^. A = 4"(16)^5'^ j^ . 

I _ . _ _ I _ T S R q I 



12, Pi2 = 412(16)'^tW? j 



388. Expandmg the first member of the equation into a series by 
division, we have 

4x- 9x2 + 30x3 -95x^ + 298x5 — 941x3 + etc. = l = - (1) 

The law of the coefficients is, that they are alternately plus and minus ; 
and that any one after the second is equal to the arithmetical sum of 
twice the preceding coefficient, plus three times the next preceding, 
plus twice the next. Eeverting the series, we have 

^^1,9 42 235 6058 247506 ^^^ 

4 a 64 a2 1024 a^ 16384 a^ 262144 a^ 4194304 a^ 

If a = 10, then x = .026363628+ = -^' 

1100 

389. We have 

^S'^ rr 1 + 16x + 63x2 + 160x3 + 325x* + 576x5 

+ ... (7i _ 2)2(3 n - 8)x^^-3 + (n - 1)2(3 n - 5)x«-2 + ^2(3 n - 2)x^-i (1) 

As there are three orders of differences in the coefficients of this series, 
we shall obtain, if we multiply (1) by (1 — x) 3, 

;S;^(l-x)3=l + 13x+18:r2(l + x+x2 + x3+...x'^-3)-(3n3+77i2 + 57i-17)x^ 
+ (6 ?22 + 5 ?i2 - 13 n + 5)x^+i - (3 n^- 2 w2)x«+2 (2) 

Summing the geometrical series in the second member of (2), and 
resolving, we obtain 

/S'n = [l + 12x+5x2-(3n3+7n2+5?i+l)x^+(9?i3+12w2-8w-r2)x^+i 

- (9 n^ + 3 ^2 - 13 ?i + 5)x^+2 + (3 # - 2 7i2)x^+3] -4- (1 - x)* 

r , . -, ,.^. 94653907(5)i'50 + 93 
= [when X = 5, and n = 100] ^ — 

128 

= 58334974030254270215112384630875216003412751797352342464364483 

02119970321656. 
When X = .999, and n = go, our general formula becomes 

S = 5x'^ + 12x+ 1 _ 17998005000. 
(1 - xy 



284 TABLE BOOK AND TEST PROBLEMS. 

390. Annual interest is simple interest on the principal, and on each 
year's interest after it is due. Put P=$400, /= $441.50 - $400 = 
$41.50, and ?^ = 4 years. Let r = the required rate. The simple interest 
on the principal for 7i years is Prn. The simple interest on one year's 
interest of the principal for 7i — 1 years is Fr^(n — 1); for n — 2 years, 
Pr^(n — 2), etc. ; the interest on the last year's interest being simply Pr^. 
Hence we have 

1= Prn + Pr2[(n - 1) + (^ - 2) + ... 1] 
= [by summing the series within brackets, etc.] Pr\_n + J ii^r — J nr'\ (1) 
Resolving (1) for r, we get 

r = [^{Phi'^ + 2 P7n2 - 2 Pin) - Pn']-^Pn{n - 1) = .025. 
Therefore the required rate is 2J per cent. 

391. Put a = $ 1000, r = .06, n = 10 years, and t = 20 years. Let P = 
the present value of the annuity a as a perpetuity ; Pj = the present value 
of the same for n years ; P^ = the present value of the same for t years, 
in reversion n years; and P^ = the present value of the perpetuity 
deferred (n + t) years. 

We evidently have P = -• The amount of an annuity of $ a for n 

years and rate r is a + a(l + 7-) -f a(l + ry + •«• a(l + r)''-^ The 
amount of its present value Pj for the same time is Pi(l + r)". 

Therefore, a + a(l + r) + a(l + ry + ... a(l + r)^-i = P^(l + r^ (1) 
Summing the geometrical series in the first member of (1), and resolv- 
ing, we get 

P.=^fl ^ ] = $7360.085, the son's bequest. 

Similarly and evidently, 
a/-. 1 



Po = - 1 - Pi = 1 = $6404.746, 

r\ ^l + ry+'J ' r(l + r)"V . (1 + OV 

the daughter's bequest. 
Finally, 

P.=P- P,- p,= ^ = $2901.835, the institution's bequest. 

' ' ' 7.(H-r)"+« 

392. Let m = the number of balls in length of the ?'th course, and n — 
the number in width of same. We shall have 

mn = h, (1); and {m + r — q){n -\- r — q) = c, (2). 
Prom (1) and (2) we obtain, by resolution, 
m and 71 = {c - & - (r - qy ± V[(c - ^)^ - 2(c + b) (r - qy -f (r - qy]} 
-- 2(r- q)= 8 and 4. 



SOLUTIONS TO SEBIES. 285 

Therefore the number of balls in the length and breadth of the bottom 
course in the pile is ?7z + ?- — 1 and n -\- r — 1 respectively ; whence, by- 
adding the number of balls in the several courses, we have the following 
series for the number in the pile : — 

(m + r - 1) (w + r - 1) + (^m + r - 2) (n + r - 2) 

+ (?7i + r - 3) (w + r-3) + ••• (m - n + 2; x 2 + (m - n + 1) x 1, 

in which ?z + r — 1 is the number of terms. Developing the terms in 
this series and summing those that are serial, we obtain for 5, the num- 
ber of balls in the pile, 

s = l(n-\-r) (n + r - 1) (3 m ~ ?i + 2 r - 1) 
= [when n = 4, m — 8, and r = 13] 2040 balls, Ans, 

393. Keeping trace of the pairs of doves produced at the end of the 
first, second, third, fourth, etc., years, we easily perceive that the num- 
bers would have been 0, 2, 3, 4, 14, 20, 44, 96, etc.. respectively. We 
therefore have to sum this series to n = 100 terms, and add to the result 
the parent pair, to find the number of pairs /Sn + 1 required in the 
problem. 

Making x -\- y + z the scale of relation in the series, we have 

Sx + 2y = 4. (1) 

ix + Sy -j-2z = U (2) 

Ux-^iy + Sz = 20 (3) 

Resolving (1), (2), and (3), we find that 

x = 0^ y = 2^ and z = i. 

Assume that the series is equal to the sum of the following three geo- 
metrical series : 

a + ar -\- ar^ + etc. (4) 

b-\-bp+ bp^ + etc. (5) 

e + eg + cg2 + etc. (6) 

that is, to 

(a + & + c) + (ar + 6i3 + eg; + (ar2 4- &i)^+cg2) + (ar3+?>J33+cg3) + etc. (7) 
The scale of relation in (7) , being 

(r + p + g) - (rp + rq + pq) + rpq, 
must be the same as that of the original series. Hence we have 

r + i) + g = ifi) 

rp + rq + pq = - 2 (9) 

and rpq = 4 (10) 



286 



TABLE BOOK AND TEST PBOBLEMS. 



Therefore the values of r, p, and q are the roots of the cubic equation 

X3_2x = 4 (11) 

that is, r = 2, j9 = - (1 + V^^), and ^ =:= - (l _ V^^). 

Hence, to make (7) identical with the original series, we must have 

a + 5 + c = (12) 

2a-(l -V^)6-(H-\/^n:)c = 2 (13) 

4a +(1 -V31)2^^(l_l_Viri)2c^3 (14) 

Resolving (12), (13), and (14), we find that 

a = J7_^ ^ =^ _ (7 + VZTl) ^ 20, and c = - (7 - V^H^) - 20. 

Substituting the numerical values found for a, &, c, r, p, and g in (4) 
(6), and (6), we obtain 

^^ 4- 1 = 1+ (0 + 2 + 3 + 4 + 14 + 20 + 44 + 96 + etc.) 

[1 + 2 + 4 4- 8 + 16 + 32 + 64 + 128 + 256 + etc.] 

[(7 + V^l)- 20][1 -(1 + V=l) + (1 + V^l)'2 

= 1 + ^ -(1 + v^n:)3 + etc.] 

[(7 _ V"=l)- 20][i -(1 - v:il) + (i - V^T)2 

_ (1 _ V-^l)3 + etc.] 

= [by summing the three geometrical series within brackets to n 
terms, and multiplying each by their respective coefficients, etc.] 

7 + V^^ ,, (- i)'^(i + \A ri)'^ - 1 



10 



l + xU2^-l)- 



20 



1,, (-1)H1 + 



20 



_2-v^n: 

(_l)n(l _vCri)n_l 
_2-f V^ 



100 



=: [by simplification when n = 100, observing that 
(_ 1)100(1 4. ^ZTiyoo = (_ 1)100(1 _ V^l) 

7 (2)100 - 3 (2)50 
10 
= 887355420159760243277720190976 pairs of doves, Ans, 

12 02 32 
394. The series — | 1 h ••• is summed as follows : — 

II II li 
The general term of the series is 

n'^ _ 7i(n + l)-(n + 1)+ 1 _ 1 _ 1 _i 1 

\n±A~ |n+ 1 ~\ n- I \)i 1 7fc + 1 



(2)50] 



SOLUTIONS TO SERIES. 287 

Hence, decomposing each term in the series, beginning with the second, 
into three parts by the aid of the formula obtained from the general term, 
we have 

12 22 32 1^111 



-I — 4- — + — H- — H y + — + - + — + 

1|2 13 14 15 ^-i-.^-1-..-t-.^-i- 



i + i + i 

1_3 [4 |5_ -^ lA ii 1^ 



= [by the exponential theorem] 

J + e-l-(^-2)+e-2-i = e -1 (1) 

32 42 52 
The series — i 1 — is summed as follows : — 

Let s = the sum of the series ; then, subtracting series (1) from this 
series, term by term, we obtain 



s — (e - 


_1)=8 12 16 ._ 4 4 4 

\1\±\± |2[3|4 




= 4(1+1 + 1 + 1 + ...^ 




= [by the exponential theorem] 4 e — 4. 


Therefore 


"ttt--^'-' 



(2) 



Adding (1) and (2), we have 



i^%^-^ + ^+-=6(.-l),^n. (3) 

In formula (3), e is the base of the Napierian system of logarithms, and 
has been computed accurately to 200 decimal places. Hence we are able 
to give the summation of the infinite series in this problem correct to that 
number of places, which will be 6(e-l)= 10.309690970754271412161724 
8281159749865434825621997574498018057663444597821212855674282930711 
5099856456479835159201835953090448157977426143740200577156357378442 
879397176566094457940297928448924246038296 +. 



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